## 数学代写|常微分方程代写ordinary differential equation代考|MATH2410

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|Basic Notions Definitions

A first-order differential equation is said to be linear if and only if it can be written as
$$\frac{d y}{d x}=f(x)-p(x) y$$
or, equivalently, as
$$\frac{d y}{d x}+p(x) y=f(x)$$
where $p(x)$ and $f(x)$ are known functions of $x$ only.
Equation (5.2) is normally considered to be the standard form for first-order linear equations. Note that the only appearance of $y$ in a linear equation (other than in the derivative) is in a term where $y$ alone is multiplied by some formula of $x$. If there are any other functions of $y$ appearing in the equation after you’ve isolated the derivative, then the equation is not linear.
Example 5.1: Consider the differential equation
$$x \frac{d y}{d x}+4 y-x^3=0 .$$

Solving for the derivative, we get
$$\frac{d y}{d x}=\frac{x^3-4 y}{x}=x^2-\frac{4}{x} y,$$
which is
$$\frac{d y}{d x}=f(x)-p(x) y$$
with
$$p(x)=\frac{4}{x} \quad \text { and } \quad f(x)=x^2 .$$
So this first-order differential equation is linear. Adding $4 / x \cdot y$ to both sides, we then get the equation in standard form,
$$\frac{d y}{d x}+\frac{4}{x} y=x^2$$
On the other hand
$$\frac{d y}{d x}+\frac{4}{x} y^2=x^2$$
is not linear because of the $y^2$.
In testing whether a given first-order differential equation is linear, it does not matter whether you attempt to rewrite the equation as
$$\frac{d y}{d x}=f(x)-p(x) y$$
or as
$$\frac{d y}{d x}+p(x) y=f(x) .$$

## 数学代写|常微分方程代写ordinary differential equation代考|Deriving the Trick for Solving

Suppose we want to solve some first-order linear equation
$$\frac{d y}{d x}+p y=f$$
(for brevity, $p=p(x)$ and $f=f(x)$ ). To avoid triviality, let’s assume $p(x)$ is not always 0 . Whether $f(x)$ vanishes or not will not be relevant.

The small trick to solving equation (5.3) comes from the product rule for derivatives: If $\mu$ and $y$ are two functions of $x$, then
$$\frac{d}{d x}[\mu y]=\frac{d \mu}{d x} y+\mu \frac{d y}{d x} .$$
Rearranging the terms on the right side, we get
$$\frac{d}{d x}[\mu y]=\mu \frac{d y}{d x}+\frac{d \mu}{d x} y,$$
and the right side of this equation looks a little like the left side of equation (5.3). To get a better match, let’s multiply equation (5.3) by $\mu$,
$$\mu \frac{d y}{d x}+\mu p y=\mu f .$$
With luck, the left side of this equation will match the right side of the last equation for the product rule, and we will have
\begin{aligned} \frac{d}{d x}[\mu y] & =\mu \frac{d y}{d x}+\frac{d \mu}{d x} y \ & =\mu \frac{d y}{d x}+\mu p y=\mu f \end{aligned}
This, of course, requires that
$$\frac{d \mu}{d x}=\mu p$$
Assuming this requirement is met, the equations in (5.4) hold. Cutting out the middle of that (and recalling that $f$ and $\mu$ are functions of $x$ only), we see that the differential equation reduces to
$$\frac{d}{d x}[\mu y]=\mu(x) f(x)$$
The advantage of having our differential equation in this form is that we can actually integrate both sides with respect to $x$, with the left side being especially easy since it is just a derivative with respect to $x$

The function $\mu$ is called an integrating factor for the differential equation. As noted in the derivation, it must satisfy
$$\frac{d \mu}{d x}=\mu p$$

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Basic Notions Definitions

$$\frac{d y}{d x}=f(x)-p(x) y$$

$$\frac{d y}{d x}+p(x) y=f(x)$$

$$x \frac{d y}{d x}+4 y-x^3=0 .$$

$$\frac{d y}{d x}=\frac{x^3-4 y}{x}=x^2-\frac{4}{x} y$$

$$\frac{d y}{d x}=f(x)-p(x) y$$

$$p(x)=\frac{4}{x} \quad \text { and } \quad f(x)=x^2$$

$$\frac{d y}{d x}+\frac{4}{x} y=x^2$$

$$\frac{d y}{d x}+\frac{4}{x} y^2=x^2$$

$$\frac{d y}{d x}=f(x)-p(x) y$$

$$\frac{d y}{d x}+p(x) y=f(x)$$

## 数学代写|常微分方程代写ordinary differential equation代考|Deriving the Trick for Solving

$$\frac{d y}{d x}+p y=f$$
(为简洁起见， $p=p(x)$ 和 $f=f(x)$ ). 为了避免琐碎，让我们假设 $p(x)$ 并不总是 0 。无论 $f(x)$ 消失与 否无关紧要。

$$\frac{d}{d x}[\mu y]=\frac{d \mu}{d x} y+\mu \frac{d y}{d x} .$$

$$\frac{d}{d x}[\mu y]=\mu \frac{d y}{d x}+\frac{d \mu}{d x} y$$

$$\mu \frac{d y}{d x}+\mu p y=\mu f$$

$$\frac{d}{d x}[\mu y]=\mu \frac{d y}{d x}+\frac{d \mu}{d x} y \quad=\mu \frac{d y}{d x}+\mu p y=\mu f$$

$$\frac{d \mu}{d x}=\mu p$$

$$\frac{d}{d x}[\mu y]=\mu(x) f(x)$$

$$\frac{d \mu}{d x}=\mu p$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH3331

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|Using and Graphing Implicit Solutions

Outside of courses specifically geared towards learning about differential equations, the main reason to solve an initial-value problem such as
$$\frac{d y}{d x}=\frac{x+1}{8+2 \pi \sin (y \pi)} \quad \text { with } \quad y(0)=2$$
is so that we can predict what values $y(x)$ will assume when $x$ has values other than 0 . In practice, of course, $y(x)$ will represent something of interest (position, velocity, promises made, number of ducks, etc.) that varies with whatever $x$ represents (time, position, money invested, food available, etc.). When the solution $y$ is given explicitly by some formula $y(x)$, then those values are relatively easily obtained by just computing that formula for different values of $x$, and a picture of how $y(x)$ varies with $x$ is easily obtained by graphing $y=y(x)$. If, instead, the solution is given implicitly by some equation, then the possible values of $y(x)$ for different $x$ ‘s, along with any graph of $y(x)$, must be extracted from that equation. It may be necessary to use advanced numerical methods to extract the desired information, but that should not be a significant problem – these methods are probably already incorporated into your favorite computer math package.
Example 4.10: Let’s consider the initial-value problem
$$\frac{d y}{d x}=\frac{x+1}{8+2 \pi \sin (y \pi)} \quad \text { with } \quad y(0)=2 \text {. }$$
In Example 4.8, we saw that the general solution to the differential equation is given implicitly by
$$8 y-2 \cos (y \pi)=\frac{1}{2} x^2+x+c .$$
The initial condition $y(0)=2$ tells us that $y=2$ when $x=0$. With this assumed, our implicit solution reduces to
$$8 \cdot 2-2 \cos (2 \pi)=\frac{1}{2}\left[0^2\right]+0+c .$$

So
$$c=8 \cdot 2-2 \cos (2 \pi)-\frac{1}{2}\left[0^2\right]-0=16-2=14 .$$
Plugging this back into equation (4.13) gives
$$8 y-2 \cos (y \pi)=\frac{1}{2} x^2+x+14$$
as an implicit solution for our initial-value problem.

## 数学代写|常微分方程代写ordinary differential equation代考|On Using Definite Integrals with Separable Equations

Just as with any directly integrable differential equation, a separable differential equation
$$\frac{d y}{d x}=f(x) g(y)$$
once separated to the form
$$\frac{1}{g(y)} \frac{d y}{d x}=f(x)$$
can be integrated using definite integrals instead of the indefinite integrals we’ve been using. The basic ideas are pretty much the same as for directly integrable differential equations:

1. Pick a convenient value for the lower limit of integration, $a$. In particular, if the value of $y\left(x_0\right)$ is given for some point $x_0$, set $a=x_0$.
2. Rewrite the differential equation with $s$ denoting the variable instead of $x$. This means that we rewrite our separable equation as
$$\frac{d y}{d s}=f(s) g(y)$$
which ‘separates’ to
$$\frac{1}{g(y)} \frac{d y}{d s}=f(s) .$$
3. Then integrate each side with respect to $s$ from $s=a$ to $s=x$.
The integral on the left-hand side will be of the form
$$\int_{s=a}^x \frac{1}{g(y)} \frac{d y}{d s} d s$$
Keep in mind that, here, $y$ is some unknown function of $s$, and that the limits in the integral are limits on $s$. Using the substitution $y=y(s)$, we see that
$$\int_{s=a}^x \frac{1}{g(y)} \frac{d y}{d s} d s=\int_{y=y(a)}^{y(x)} \frac{1}{g(y)} d y .$$
Do not forget to convert the limits to being the corresponding limits on $y$, instead of $s$.
Once the integration is done, we attempt to solve the resulting equation for $y(x)$ just as before.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Using and Graphing Implicit Solutions

$$\frac{d y}{d x}=\frac{x+1}{8+2 \pi \sin (y \pi)} \quad \text { with } \quad y(0)=2$$

$$\frac{d y}{d x}=\frac{x+1}{8+2 \pi \sin (y \pi)} \quad \text { with } \quad y(0)=2 .$$

$$8 y-2 \cos (y \pi)=\frac{1}{2} x^2+x+c .$$

$$8 \cdot 2-2 \cos (2 \pi)=\frac{1}{2}\left[0^2\right]+0+c .$$

$$c=8 \cdot 2-2 \cos (2 \pi)-\frac{1}{2}\left[0^2\right]-0=16-2=14 .$$

$$8 y-2 \cos (y \pi)=\frac{1}{2} x^2+x+14$$

## 数学代写|常微分方程代写ordinary differential equation代考|On Using Definite Integrals with Separable Equations

$$\frac{d y}{d x}=f(x) g(y)$$

$$\frac{1}{g(y)} \frac{d y}{d x}=f(x)$$

1. 为积分的下限选择一个方便的值， $a$. 特别是，如果值 $y\left(x_0\right)$ 给出了一些点 $x_0$ ，放 $a=x_0$. 2. 重写微分方程 $s$ 表示变量而不是 $x$. 这意味着我们将可分离方程重写为
$$\frac{d y}{d s}=f(s) g(y)$$
哪个”分开”到
$$\frac{1}{g(y)} \frac{d y}{d s}=f(s)$$
2. 然后整合每一方 $s$ 从 $s=a$ 到 $s=x$. 左侧的积分形式为
$$\int_{s=a}^x \frac{1}{g(y)} \frac{d y}{d s} d s$$
请记住，在这里， $y$ 是一些末知的功能 $s$ ，并且积分中的限制是限制 $s$. 使用替换 $y=y(s)$, 我们看到
$$\int_{s=a}^x \frac{1}{g(y)} \frac{d y}{d s} d s=\int_{y=y(a)}^{y(x)} \frac{1}{g(y)} d y .$$
不要忘记将限制转换为相应的限制 $y$ ，代替 $s$.
积分完成后，我们尝试求解结果方程 $y(x)$ 就像以前一样。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH53

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|A Caution on False Solutions

It is always a good idea to verify that any ‘solution’ obtained in solving a differential equation really is a solution. This is even more true when solving separable differential equations. Not only does the extra algebra involved naturally increase the likelihood of human error, this algebra can, as noted above, lead to ‘false solutions’ – formulas that are obtained as solutions but do not actually satisfy the original problem.
Example 4.9: Consider the initial-value problem
$$\frac{d y}{d x}=2 \sqrt{y} \quad \text { with } \quad y(0)=4 \quad \text {. }$$
The differential equation does have one constant solution, $y=0$, but since that doesn’t satisfy the initial condition, it hardly seems relevant. To find the other solutions, let’s divide the differential equation by $\sqrt{y}$ and proceed with the basic procedure:
$$\begin{gathered} \frac{1}{\sqrt{y}} \frac{d y}{d x}=2 \ \int \frac{1}{\sqrt{y}} \frac{d y}{d x} d x=\int 2 d x \end{gathered}$$
$$\hookrightarrow$$ \begin{aligned} & \leftrightarrow \quad \int y^{-1 / 2} d y=\int 2 d x \ & \hookrightarrow \quad 2 y^{1 / 2}=2 x+c . \ & \end{aligned}
Dividing by 2 and squaring (and letling $a=c / 2$ ), we get
$$y=(x+a)^2 .$$
Plugging this into the initial condition, we obtain
$$4=y(0)=(0+a)^2=a^2$$
which means that
$$a=\pm 2$$
Hence, we have two formulas for the solution to our initial-value problem,
$$y_{+}(x)=(x+2)^2 \quad \text { and } \quad y_{-}(x)=(x-2)^2 .$$
Both satisfy the initial condition. Do both satisfy the differential equation
$$\frac{d y}{d x}=2 \sqrt{y} \quad ?$$
Well, plugging
$$y=y_{\pm}(x)=(x \pm 2)^2$$
into the differential equation yields
\begin{aligned} \frac{d}{d x}(x \pm 2)^2 & =2 \sqrt{(x \pm 2)^2} \ \longleftrightarrow \quad 2(x \pm 2) & =2 \sqrt{(x \pm 2)^2} \end{aligned}

## 数学代写|常微分方程代写ordinary differential equation代考|On the Nature of Solutions to Differential Equations

When we solve a first-order directly integrable differential equation,
$$\frac{d y}{d x}=f(x)$$
we get something of the form
$$y=F(x)+c$$
where $F$ is any antiderivative of $f$ and $c$ is an arbitrary constant. Computationally, all we have to do is find a single antiderivative $F$ for $f$ and then add an arbitrary constant. Thus, also, the graph of any possible solution is nothing more than the graph of $F(x)$ shifted vertically by the value of $c$ (up if $c>0$, down if $c<0$ ). What’s more, the interval for $x$ over which
$$y=F(x)+c$$
is a valid solution depends only on the one function $F$. If $F(x)$ is continuous for all $x$ in an interval $(a, b)$, then $(a, b)$ is a valid interval for our solution. This interval does not depend on the choice for $c$.

The situation can be much more complicated if our differential equation is not directly integrable. First of all, finding an explicit solution can be impossible. And consider those explicit general solutions we have found,
$$y=\tan \left(\frac{1}{3} x^3+c\right) \quad \text { (from Example } 4.3 \text { on page 67) }$$
and
$$y=3 \pm \sqrt{a-x^2} \quad \text { (from Example } 4.4 \text { on page 69) . }$$
In both of these, the arbitrary constants are not simply “added on” to some formula of $x$. Instead, each solution formula combines the variable, $x$, with the arbitrary constant, $c$ or $a$, in a very nontrivial manner. There are two immediate consequences of this:

1. The graphs of the solutions are no longer simply vertically shifted copies of some single function.
2. The possible intervals over which any solution is valid may depend on the arbitrary constant. And since the value of that constant can be determined by the initial condition, the interval of validity for our solutions may depend on the initial condition.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|A Caution on False Solutions

$$\frac{d y}{d x}=2 \sqrt{y} \quad \text { with } \quad y(0)=4$$

\begin{aligned} \frac{1}{\sqrt{y}} \frac{d y}{d x} & =2 \int \frac{1}{\sqrt{y}} \frac{d y}{d x} d x=\int 2 d x \ \leftrightarrow & \hookrightarrow \ \leftrightarrow y^{-1 / 2} d y & =\int 2 d x \quad 2 y^{1 / 2}=2 x+c . \end{aligned}

$$y=(x+a)^2 .$$

$$4=y(0)=(0+a)^2=a^2$$

$$a=\pm 2$$

$$y_{+}(x)=(x+2)^2 \quad \text { and } \quad y_{-}(x)=(x-2)^2 .$$

$$\frac{d y}{d x}=2 \sqrt{y} \quad ?$$

$$y=y_{\pm}(x)=(x \pm 2)^2$$

$$\frac{d}{d x}(x \pm 2)^2=2 \sqrt{(x \pm 2)^2} \longleftrightarrow 2(x \pm 2) \quad=2 \sqrt{(x \pm 2)^2}$$

## 数学代写|常微分方程代写ordinary differential equation代考|On the Nature of Solutions to Differential Equations

$$\frac{d y}{d x}=f(x)$$

$$y=F(x)+c$$

$$y=F(x)+c$$

$$y=\tan \left(\frac{1}{3} x^3+c\right) \quad \text { (from Example } 4.3 \text { on page } 67 \text { ) }$$

$$\left.y=3 \pm \sqrt{a-x^2} \quad \text { (from Example } 4.4 \text { on page } 69\right) .$$

1. 解决方案的图形不再是某些单个函数的简单垂直移动副本。
2. 任何解决方案有效的可能间隔可能取决于任意常数。由于该常数的值可以由初始条件确定，因此我 们解决方案的有效性区间可能取决于初始条件。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH211

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|The Simplest Falling Object Model

The Earth’s gravity is the most obvious force acting on our falling object. Checking a convenient physics text, we find that the force of the Earth’s gravity acting on an object of mass $m$ is given by
$$F_{\text {grav }}=-g m \quad \text { where } \quad g=9.8\left(\text { meters } / \text { second }^2\right) .$$
Of course, the value for $g$ is an approximation and assumes that the object is not too far above the Earth’s surface. It also assumes that we’ve chosen “up” to be the positive direction (hence the negative sign).

For this model, let us suppose the Earth’s gravity, $F_{\text {grav }}$, is the only significant force involved. Assuming this (and keeping in mind that we are measuring distance in meters and time in seconds), we have
$$F=F_{\text {grav }}=-9.8 \mathrm{~m}$$
in the ” $F=m a$ ” equation. In particular, equation $\left(1.2^{\prime}\right)$ becomes
$$-9.8 m=m \frac{d^2 y}{d t^2} .$$
The mass conveniently divides out, leaving us with
$$\frac{d^2 y}{d t^2}=-9.8 .$$
Taking the indefinite integral with respect to $t$ of both sides of this equation yields
$$\begin{array}{rlrl} \int \frac{d^2 y}{d t^2} d t & =\int-9.8 d t \ \hookrightarrow \quad \int \frac{d}{d t}\left(\frac{d y}{d t}\right) d t & =\int-9.8 d t \ \hookrightarrow & \frac{d y}{d t}+c_1 & =-9.8 t+c_2 \ \hookrightarrow & \frac{d y}{d t} & =-9.8 t+c \end{array}$$
where $c_1$ and $c_2$ are the arbitrary constants of integration and $c=c_2-c_1$. This gives us our formula for ${ }^{d y / d t}$ up to an unknown constant $c$. But recall that the initial velocity is zero.
$$\left.\frac{d y}{d t}\right|_{t=0}=v(0)=0 .$$

## 数学代写|常微分方程代写ordinary differential equation代考|A Better Falling Object Model

The above model does not take into account the resistance of the air to the falling object – a very important force if the object is relatively light or has a parachute. Let us add this force to our model. That is, for our ” $F=m a$ ” equation, we’ll use
$$F=F_{\text {grav }}+F_{\text {air }}$$
where $F_{\text {grav }}$ is the force of gravity discussed above, and $F_{\text {air }}$ is the force due to the air resistance acting on this particular falling body.

Part of our problem now is to determine a good way of describing $F_{\text {air }}$ in terms relevant to our problem. To do that, let us list a few basic properties of air resistance that should be obvious to anyone who has stuck their hand out of a car window:

1. The force of air resistance does not depend on the position of the object, only on the relative velocity between it and the surrounding air. So, for us, $F_{\text {air }}$ will just be a function of $v$, $F_{\text {air }}=F_{\text {air }}(v)$. (This assumes, of course, that the air is still – no up-or downdrafts – and that the density of the air remains fairly constant throughout the distance this object falls.)
2. This force is zero when the object is not moving, and its magnitude increases as the speed increases (remember, speed is the magnitude of the velocity). Hence, $F_{\mathrm{air}}(v)=0$ when $v=0$, and $\left|F_{\text {air }}(v)\right|$ gets bigger as $|v|$ gets bigger.
3. Air resistance acts against the direction of motion. This means that the direction of the force of air resistance is opposite to the direction of motion. Thus, the sign of $F_{\text {air }}(v)$ will be opposite that of $v$.

While there are many formulas for $F_{\text {air }}(v)$ that would satisfy the above conditions, common sense suggests that we first use the simplest. That would be
$$F_{\mathrm{air}}(v)=-\gamma v$$ where $\gamma$ is some positive value. The actual value of $\gamma$ will depend on such parameters as the object’s size, shape, and orientation, as well as the density of the air through which the object is moving. For any given object, this value could be determined by experiment (with the aid of the equations we will soon derive).

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|The Simplest Falling Object Model

$$F=F_{\text {grav }}=-9.8 \mathrm{~m}$$

$$-9.8 m=m \frac{d^2 y}{d t^2} .$$

$$\frac{d^2 y}{d t^2}=-9.8 .$$

$$\int \frac{d^2 y}{d t^2} d t=\int-9.8 d t \hookrightarrow \int \frac{d}{d t}\left(\frac{d y}{d t}\right) d t=\int-9.8 d t \hookrightarrow \frac{d y}{d t}+c_1=-9.8 t+c_2 \hookrightarrow \frac{d y}{d t}$$

$$\left.\frac{d y}{d t}\right|_{t=0}=v(0)=0$$

## 数学代写|常微分方程代写ordinary differential equation代考|A Better Falling Object Model

$$F=F_{\text {grav }}+F_{\text {air }}$$

1. 空气阻力的大小与物体的位置无关，只与物体与周围空气的相对速度有关。所以，对我们来说， $F_{\text {air }}$ 将只是一个函数 $v, F_{\text {air }}=F_{\text {air }}(v)$. (当然，这是假设空气是静止的一一没有上升气流或下 降气流一一并且在这个物体下落的整个距离内空气的密度保持相当恒定。)
2. 当物体不动时这个力为零，它的大小随㸔速度的增加而增加（记住，速度是速度的大小）。因此， $F_{\text {air }}(v)=0$ 什么时候 $v=0$ ，和 $\left|F_{\text {air }}(v)\right|$ 变大为 $|v|$ 变大。
3. 空气阻力与运动方向相反。这意味着空气阻力的方向与运动方向相反。因此，标志 $F_{\text {air }}(v)$ 将与 $v$.
虽然有很多公式 $F_{\text {air }}(v)$ 满足上述条件，常识建议我们首先使用最简单的。那将是
$$F_{\text {air }}(v)=-\gamma v$$
在哪里 $\gamma$ 是一些正值。的实际价值 $\gamma$ 将取决于物体的大小、形状和方向等参数，以及物体移动时空气的密 度。对于任何给定的物体，这个值可以通过实验来确定（借助于我们很快就会推导出来的方程式）。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH2410

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|Initial-Value Problems

One set of auxiliary conditions that often arises in applications is a set of “initial values” for the desired solution. This is a specification of the values of the desired solution and some of its derivatives at a single point. To be precise, an $N^{\text {th }}$-order set of initial values for a solution $y$ consists of an assignment of values to
$$y\left(x_0\right) \quad, \quad y^{\prime}\left(x_0\right) \quad, \quad y^{\prime \prime}\left(x_0\right) \quad, \quad y^{\prime \prime \prime}\left(x_0\right) \quad, \quad \ldots \quad \text { and } \quad y^{(N-1)}\left(x_0\right)$$
where $x_0$ is some fixed number (in practice, $x_0$ is often 0 ) and $N$ is some nonnegative integer. ${ }^3$ Note that there are exactly $N$ values being assigned and that the highest derivative in this set is of order $N-1$.

We will find that $N^{\text {th }}$-order sets of initial values are especially appropriate for $\mathrm{N}^{\text {th }}$-order differential equations. Accordingly, the term $N^{\text {th }}$-order initial-value problem will always mean a problem consisting of

1. an $N^{\text {th }}$-order differential equation, and
2. an $N^{\text {th }}$-order set of initial values.
For example,
$$\frac{d y}{d x}-3 y=0 \quad \text { with } \quad y(0)=4$$
is a first-order initial-value problem. $” d y / d x-3 y=0 “$ is the first-order differential equation, and $” y(0)=4 “$ is the first-order set of initial valucs. On the other hand, the third-order differential equation
$$\frac{d^3 y}{d x^3}+\frac{d y}{d x}=0$$
along with the third-order set of initial conditions
$$y(1)=3 \quad, \quad y^{\prime}(1)=-4 \text { and } y^{\prime \prime}(1)=10$$
makes up a third-order initial-value problem.
A solution to an initial-value problem is a solution to the differential equation that also satisfies the given initial values. The usual approach to solving such a problem is to first find the general solution to the differential equation (via any of the methods we’ll develop later), and then determine the values of the ‘arbitrary’ constants in the general solution so that the resulting function also satisfies each of the given initial values.

## 数学代写|常微分方程代写ordinary differential equation代考|The Situation to Be Modeled

Let us concern ourselves with the vertical position and motion of an object dropped from a plane at a height of 1,000 meters. Since it’s just being dropped, we may assume its initial downward velocity is 0 meters per second. The precise nature of the object – whether it’s a falling marble, a frozen duck (live, unfrozen ducks don’t usually fall) or some other familiar falling object — is not important at this time. Visualize it as you will.

The first two things one should do when developing a model is to sketch the process (if possible) and to assign symbols to quantitics that may be relevant. A crude sketch of the process is in Figure $1.1$ (I’ve sketched the object as a ball since a ball is easy to sketch). Following ancient traditions, let’s make the following symbolic assignments:
\begin{aligned} m & =\text { the mass (in grams) of the object } \ t & =\text { time (in seconds) since the object was dropped } \end{aligned}

$y(t)=$ vertical distance (in meters) between the object and the ground at time $t$
$v(t)=$ vertical velocity (in meters/second) of the object at time $t$
$a(t)=$ vertical acceleration (in meters $/$ second $^2$ ) of the object at time $t$
Where convenient, we will use $y, v$ and $a$ as shorthand for $y(t), v(t)$ and $a(t)$. Remember that, by the definition of velocity and acceleration,
$$v=\frac{d y}{d t} \quad \text { and } \quad a=\frac{d v}{d t}=\frac{d^2 y}{d t^2} .$$
From our assumptions regarding the object’s position and velocity at the instant it was dropped, we have that
$$y(0)=1,000 \quad \text { and }\left.\quad \frac{d y}{d t}\right|_{t=0}=v(0)=0 .$$
These will be our initial values. (Notice how appropriate it is to call these the “initial values” $y(0)$ and $v(0)$ are, indeed, the initial position and velocity of the object.)

As time goes on, we expect the object to be falling faster and faster downwards, so we expect both the position and velocity to vary with time. Precisely how these quantities vary with time might be something we don’t yet know. However, from Newton’s laws, we do know
$$F=m a$$
where $F$ is the sum of the (vertically acting) forces on the object. Replacing $a$ with either the corresponding derivative of velocity or position, this equation becomes
$$F=m \frac{d v}{d t}$$
or, equivalently,
$$F=m \frac{d^2 y}{d t^2} .$$

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Initial-Value Problems

$$y\left(x_0\right) \quad, \quad y^{\prime}\left(x_0\right) \quad, \quad y^{\prime \prime}\left(x_0\right) \quad, \quad y^{\prime \prime \prime}\left(x_0\right) \quad, \quad \ldots \quad \text { and } y^{(N-1)}\left(x_0\right)$$

1. 一个 $N^{\text {th }}$ 阶溦分方程，和
2. 一个 $N^{\text {th }}$-order 集合的初始值。
例如，
$$\frac{d y}{d x}-3 y=0 \quad \text { with } \quad y(0)=4$$
是一阶初值问题。” $d y / d x-3 y=0$ “是一阶微分方程，并且” $y(0)=4$ “是一阶初始值集。另一 方面，三阶微分方程
$$\frac{d^3 y}{d x^3}+\frac{d y}{d x}=0$$
连同三阶初始条件集
$$y(1)=3 \quad, \quad y^{\prime}(1)=-4 \text { and } y^{\prime \prime}(1)=10$$
组成一个三阶初值问题。
初值问题的解是微分方程也满足给定初值的解。解决此类问题的通常方法是首先找到微分方程的通 解（通过我们稍后将开发的任何方法），然后确定通解中“任意”常数的值，以便结果函数也满足每 个给定的初始值。

## 数学代写|常微分方程代写ordinary differential equation代考|The Situation to Be Modeled

$m=$ the mass (in grams) of the object $t \quad=$ time (in seconds) since the object was
$y(t)=$ 时间物体与地面之间的垂直距离 (以米为单位) $t$
$v(t)=$ 物体当时的垂直速度 (以米/秒为单位) $t$
$a(t)=$ 垂直加速度 (米/第二 $\left.{ }^2\right)$ 对象的时间 $t$

$v=\frac{d y}{d t} \quad$ and $\quad a=\frac{d v}{d t}=\frac{d^2 y}{d t^2}$.

$$y(0)=1,000 \quad \text { and }\left.\quad \frac{d y}{d t}\right|_{t=0}=v(0)=0 .$$

$$F=m a$$

$$F=m \frac{d v}{d t}$$

$$F=m \frac{d^2 y}{d t^2}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH53

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|Basic Definitions and Classifications

A differential equation is an equation involving some function of interest along with a few of its derivatives. Typically, the function is unknown, and the challenge is to determine what that function could possibly be.

Differential equations can be classified either as “ordinary” or as “partial”. An ordinary differential equation is a differential equation in which the function in question is a function of only one variable. Hence, its derivatives are the “ordinary” derivatives encountered early in calculus. For the most part, these will be the sort of equations we’ll be examining in this text. For example,
$$\begin{gathered} \frac{d y}{d x}=4 x^3 \ \frac{d y}{d x}+\frac{4}{x} y=x^2 \ \frac{d^2 y}{d x^2}-2 \frac{d y}{d x}-3 y=65 \cos (2 x) \end{gathered}$$ and
$$4 x^2 \frac{d^2 y}{d x^2}+4 x \frac{d y}{d x}+\left[4 x^2-1\right] y=0$$
$$\frac{d^4 y}{d x^4}=81 y$$
are some differential equations that we will later deal with. In each, $y$ denotes a function that is given by some, yet unknown, formula of $x$. Of course, there is nothing sacred about our choice of symbols. We will use whatever symbols are convenient for the variables and functions, especially if the problem comes from an application and the symbols help remind us of what they denote (such as when we use $t$ for a measurement of time). ${ }^1$

A partial differential equation is a differential equation in which the function of interest depends on two or more variables. Consequently, the derivatives of this function are the partial derivatives developed in the later part of most calculus courses. ${ }^2$ Because the methods for studying partial differential equations often involve solving ordinary differential equations, it is wise to first become reasonably adept at dealing with ordinary differential equations before tackling partial differential equations.

As already noted, this text is mainly concerned with ordinary differential equations. So let us agree that, unless otherwise indicated, the phrase “differential equation” in this text means “ordinary differential equation”. If you wish to further simplify the phrasing to “DE” or even to something like “Diffy-Q”, go ahead. This author, however, will not be so informal.

## 数学代写|常微分方程代写ordinary differential equation代考|The Basic Notions

Any function that satisfies a given differential equation is called a solution to that differential equation. “Satisfies the equation”, means that, if you plug the function into the differential equation and compute the derivatives, then the result is an equation that is true no matter what real value we replace the variable with. And if that resulting equation is not true for some real values of the variable, then that function is not a solution to that differential equation.
Example 1.1: Consider the differential equation
$$\frac{d y}{d x}-3 y=0 .$$
If, in this differential equation, we let $y(x)=e^{3 x}$ (i.e., if we replace $y$ with $e^{3 x}$ ), we get
\begin{aligned} & \frac{d}{d x}\left[e^{3 x}\right]-3 e^{3 x}=0 \ & \longleftrightarrow \quad 3 e^{3 x}-3 e^{3 x}=0 \ & \longleftrightarrow \quad 0=0 \ & 0 \quad \end{aligned}
which certainly is true for every real value of $x$. So $y(x)=e^{3 x}$ is a solution to our differential equation.
On the other hand, if we let $y(x)=x^3$ in this differential equation, we get
\begin{aligned} & \frac{d}{d x}\left[x^3\right]-3 x^3=0 \ & \longrightarrow \quad 3 x^2-3 x^3=0 \ & \longleftrightarrow \quad 3 x^2(1-x)=0 \ & 3 \quad \end{aligned}
which is true only if $x=0$ or $x=1$. But our interest is not in finding values of $x$ that make the equation true; our interest is in finding functions of $x$ (i.e., $y(x)$ ) that make the equation true for all values of $x$. So $y(x)=x^3$ is not a solution to our differential equation. (And it makes no sense, whatsoever, to refer to either $x=0$ or $x=1$ as solutions, here.)

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Basic Definitions and Classifications

$$\frac{d y}{d x}=4 x^3 \frac{d y}{d x}+\frac{4}{x} y=x^2 \frac{d^2 y}{d x^2}-2 \frac{d y}{d x}-3 y=65 \cos (2 x)$$

$$\begin{gathered} 4 x^2 \frac{d^2 y}{d x^2}+4 x \frac{d y}{d x}+\left[4 x^2-1\right] y=0 \ \frac{d^4 y}{d x^4}=81 y \end{gathered}$$

## 数学代写|常微分方程代写ordinary differential equation代考|The Basic Notions

$$\frac{d y}{d x}-3 y=0$$

$$\frac{d}{d x}\left[e^{3 x}\right]-3 e^{3 x}=0 \quad \longleftrightarrow 3 e^{3 x}-3 e^{3 x}=0 \longleftrightarrow 0=0 \quad 0$$

$$\frac{d}{d x}\left[x^3\right]-3 x^3=0 \quad \longrightarrow 3 x^2-3 x^3=0 \longleftrightarrow 3 x^2(1-x)=0 \quad 3$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH2410

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|Linear ODEs

Another important type of ODE which can be solved easily is the linear equation (both homogeneous and non-homogeneous). Let $J$ be a closed interval and $P: J \rightarrow \mathbb{R}$ be a continuous function. An equation of the form
$$y^{\prime}(x)+P(x) y(x)=0$$
is called a first order linear homogeneous ODE. If $Q$ is a nonzero continuous function on $J$, then
$$y^{\prime}(x)+P(x) y(x)=Q(x)$$
is called a first order linear non-homogeneous ODE. Any first order ODE that we consider in this chapter which is not in any of the forms (2.26) or (2.27) is called a nonlinear $O D E$.

There are many ways to solve (2.26). One of them is to apply the method of separation of variables. On comparing (2.26) with (2.1), we get
$$f(x)=-P(x), g(y)=\frac{1}{y} .$$
Therefore a solution to (2.26) is implicitly given by
$$\begin{gathered} \int^y \frac{d y}{y}=-\int^x P(x) d x+\tilde{c}, \tilde{c} \in \mathbb{R}, \ y=e^{\tilde{c}} e^{-\int^x P(x) d x} . \end{gathered}$$
From the previous relation, we directly obtain that
$$\phi(x)=c e^{-\int^x P(x) d x}, c \in \mathbb{R},$$
is a solution to (2.26). We now describe another way of obtaining the solution given in (2.28). Let $\phi$ be a solution to (2.26). On substituting $\phi$ in (2.26) and multiplying with $e^{\int^x P(x) d x}$ on both sides, we arrive at
or
$$\begin{gathered} e^{\int^x P(x) d x} \frac{d \phi(x)}{d x}+\frac{d}{d x}\left(e^{\int^x P(x) d x}\right) \phi(x)=0 \ \frac{d}{d x}\left(\phi(x) e^{\int^x P(x) d x}\right)=0 \end{gathered}$$

## 数学代写|常微分方程代写ordinary differential equation代考|Well-posedness

Throughout this chapter, we assume that every interval that we consider has a positive length ${ }^3$. We assume that $J$ and $\Omega$ are open intervals in $\mathbb{R}$. Let $\bar{J}$ and $\bar{\Omega}$ denote the smallest closed intervals containing $J$ and $\Omega$, respectively. Let $f: \bar{J} \times \bar{\Omega} \rightarrow \mathbb{R}$ be a function. Consider the problem
$$\left{\begin{array}{l} y^{\prime}(x)=f(x, y(x)), x \in J, \ y\left(x_0\right)=y_0 . \end{array}\right.$$
Definition 2.2.1. Let $J_1 \subseteq \bar{J}$ be an interval containing $x_0$. We say that a function $\phi: J_1 \rightarrow \mathbb{R}$ is said to be a solution to (2.34) if
(i) $\phi \in C\left(J_1\right) \cap C^1\left(J_1^o\right)$, where $J_1^o$ is the interval (inf $J_1, \sup J_1$ ),
(ii) $\phi(x) \in \Omega, x \in J_1$,
(iii) on substituting $y=\phi$ in (2.34) we get an identity in $J_1$.
Moreover, if $J_1 \backslash\left{x_0\right} \subset J \backslash\left{x_0\right}$, then we say that $\phi$ is a local solution. Otherwise it is called a global solution. If $J_1$ is of the form $\left[x_0, x_1\right]$ or $\left[x_0, x_1\right)$, then we say that $\phi$ is a right solution. If $J_1$ is of the form $\left[x_1, x_0\right]$ or $\left(x_1, x_0\right]$, then we say that $\phi$ is a left solution. If $x_0 \in J_1^o$ then we say that $\phi$ is a bilateral solution. If $J=\left(x_0, x_1\right)$ where $x_1 \in \mathbb{R} \cup{\infty}$, then (2.34) is said to be an initial value problem (IVP) and we deal with the right solutions in the study of IVPs. On the other hand, if $x_0 \in J$ then (2.34) is said to be a Cauchy problem. We usually seek bilateral solutions while studying Cauchy problems.
In fact, one of the main theorems of this chapter is to prove the existence of a bilateral (right) solutions to Cauchy problems (IVPs).

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Linear ODEs

$$y^{\prime}(x)+P(x) y(x)=0$$

$$y^{\prime}(x)+P(x) y(x)=Q(x)$$

(2.26)有多种求解方法。其中之一是应用变量分离法。将 (2.26) 与 (2.1) 进行比较，我们得到
$$f(x)=-P(x), g(y)=\frac{1}{y}$$

$$\int^y \frac{d y}{y}=-\int^x P(x) d x+\tilde{c}, \tilde{c} \in \mathbb{R}, y=e^{\bar{c}} e^{-\int^x P(x) d x}$$

$$\phi(x)=c e^{-\int^x P(x) d x}, c \in \mathbb{R}$$

$$e^{\int^x P(x) d x} \frac{d \phi(x)}{d x}+\frac{d}{d x}\left(e^{f^x P(x) d x}\right) \phi(x)=0 \frac{d}{d x}\left(\phi(x) e^{f^x P(x) d x}\right)=0$$

## 数学代写|常微分方程代写ordinary differential equation代考|Well-posedness

y^{\prime}(x)=f(x, y(x)), x \in J, y\left(x_0\right)=y_0
$$正确的。 \ \$$

(二) $\phi(x) \in \Omega, x \in J_1$,
(iii) 关于替代 $y=\phi$ 在 (2.34) 中我们得到一个恒等式 $J_1$. 决方案。如果 $J_1$ 是形式 $\left[x_0, x_1\right]$ 要么 $\left[x_0, x_1\right)$ ，那么我们说 $\phi$ 是一个正确的解决方案。如果 $J_1$ 是形式 $\left[x_1, x_0\right]$ 要么 $\left(x_1, x_0\right]$ ，那么我们说 $\phi$ 是左解。如果 $x_0 \in J_1^o$ 然后我们说 $\phi$ 是双边解决方案。如果
$J=\left(x_0, x_1\right)$ 在哪里 $x_1 \in \mathbb{R} \cup \infty$ ，那么 (2.34) 被称为初始值问题 (IVP) 并且我们在 IVP 的研究中处理正 确的解决方案。另一方面，如果 $x_0 \in J$ 则 (2.34) 被称为柯西问题。我们在研究柯西问题时通常寻求双边 解快方案。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH3331

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|Separation of variables

Consider the ODE of the form
$$\frac{d}{d x} y(x)=\frac{f(x)}{g(y(x))} .$$
We assume that $f:\left(a_0, a_1\right) \rightarrow \mathbb{R}$ and $g:\left(b_0, b_1\right) \rightarrow(0, \infty)$ are continuous functions. Wè also assume that there exists $y_0$ in the interval $\left(b_0, b_1\right)$ such that
$$g\left(y_0\right) \neq 0 .$$
We define a function $F:\left(a_0, a_1\right) \times\left(b_0, b_1\right) \rightarrow \mathbb{R}$ by
$$F(x, y)=\int_{y_0}^y g(\xi) d \xi-\int_{x_0}^x f(s) d s, x \in\left(a_0, a_1\right), y \in\left(b_0, b_1\right) .$$
Since $f$ and $g$ are continuous, $F$ is a $C^1$-function. Moreover for every $x_0 \in$ $\left(a_0, a_1\right)$ we have
$$\frac{\partial F}{\partial y}\left(x_0, y_0\right)=g\left(y_0\right) \neq 0 \text {. }$$

Therefore by the implicit function theorem (see Appendix C) there exists $\delta>0$ and a $C^1$-function $\phi:\left(x_0-\delta, x_0+\delta\right) \rightarrow \mathbb{R}$ such that
$$F(x, \phi(x))=\int_{y_0}^{\phi(x)} g(\xi) d \xi-\int_{x_0}^x f(s) d s=F\left(x_0, y_0\right), x \in\left(x_0-\delta, x_0+\delta\right) .$$
One can easily prove that $\phi$ is a solution to (2.1). For, on differentiating (2.3) with respect to $x$ (using the Leibniz rule of differentiation ${ }^1$ ) we get
$$\phi^{\prime}(x) g(\phi(x))-f(x)=0, x \in\left(x_0-\delta, x_0+\delta\right) .$$
This proves that the function $\phi$ which is implicitly given by the relation $F(x, y)=F\left(x_0, y_0\right)$, is a solution to (2.1). In other words, the relation
$$\int^y g(y) d y=\int^x f(x) d x+c, c \in \mathbb{R},$$
where the above integrals are indefinite integrals, defines a solution to (2.1). We now present some examples where this technique is demonstrated.

## 数学代写|常微分方程代写ordinary differential equation代考|Exact cquations

In this subsection, we present another special form of differential equations called exact equations which can be solved easily. Let $M, N$ be continuous functions in a rectangle
$$R=\left{(x, y):\left|x-x_0\right| \leq a,\left|y-y_0\right| \leq b\right},$$
and $N$ does not vanish in $R$. An ODE of the form
$$N(x, y(x)) y^{\prime}(x)+M(x, y(x))=0,$$
is said to be exact if there exists a $C^1$-function $F: R \rightarrow \mathbb{R}$ such that
$$\frac{\partial F}{\partial x}(x, y)=M(x, y), \quad \frac{\partial F}{\partial y}(x, y)=N(x, y),(x, y) \in R .$$
Example 2.1.8. Show that $y(x) y^{\prime}(x)+x=0$ is an exact equation.
Solution. In order to prove this, we first compare the given equation with (2.18) to get $M(x, y)=x$ and $N(x, y)=y$. It is easy to verify that

$$F(x, y)=\frac{x^2+y^2}{2},$$
satisfies (2.19). Hence the given equation is exact.
We now establish the connection between $F$ and the solutions to (2.18). To this end, we suppose (2.18) is exact and $F$ is known to us. We observe that $\frac{\partial F}{\partial y}=N \neq 0$, in $R$. Let $(\tilde{x}, \tilde{y}) \in \mathbb{R}^2$ satisfy $\left|x_0-\tilde{x}\right|<a$ and $\left|y_0-\tilde{y}\right|<b$. Then by the implicit function theorem there exists an interval $(\tilde{x}-\delta, \tilde{x}+\delta)$, which is denoted by $J$, and a $C^1$-function $\phi: J \rightarrow \mathbb{R}$ such that
$$F(x, \phi(x))=F(\tilde{x}, \tilde{y}), x \in J .$$
Claim. The function $\phi$ is a solution to (2.18).
For, on differentiating (2.20) with respect to $x$ we get
$$\frac{\partial F}{\partial x}(x, \phi(x))+\frac{\partial F}{\partial y}(x, \phi(x)) \phi^{\prime}(x)=0, x \in J .$$
Thus we have
$$M(x, \phi(x))+N(x, \phi(x)) \phi^{\prime}(x)=0, x \in J,$$
which proves that $\phi$ is a solution to (2.18). Hence the claim is proved.
Now, we shall revisit Example 2.1.8 and solve the ODE therein.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Separation of variables

$$\frac{d}{d x} y(x)=\frac{f(x)}{g(y(x))}$$

$$g\left(y_0\right) \neq 0 .$$

$$F(x, y)=\int_{y_0}^y g(\xi) d \xi-\int_{x_0}^x f(s) d s, x \in\left(a_0, a_1\right), y \in\left(b_0, b_1\right) .$$

$$\frac{\partial F}{\partial y}\left(x_0, y_0\right)=g\left(y_0\right) \neq 0 .$$

$$F(x, \phi(x))=\int_{y_0}^{\phi(x)} g(\xi) d \xi-\int_{x_0}^x f(s) d s=F\left(x_0, y_0\right), x \in\left(x_0-\delta, x_0+\delta\right)$$

$$\phi^{\prime}(x) g(\phi(x))-f(x)=0, x \in\left(x_0-\delta, x_0+\delta\right) .$$

$$\int^y g(y) d y=\int^x f(x) d x+c, c \in \mathbb{R}$$

## 数学代写|常微分方程代写ordinary differential equation代考|Exact cquations

$$N(x, y(x)) y^{\prime}(x)+M(x, y(x))=0,$$

$$\frac{\partial F}{\partial x}(x, y)=M(x, y), \quad \frac{\partial F}{\partial y}(x, y)=N(x, y),(x, y) \in R .$$

$$F(x, y)=\frac{x^2+y^2}{2}$$

$$F(x, \phi(x))=F(\tilde{x}, \tilde{y}), x \in J .$$

$$\frac{\partial F}{\partial x}(x, \phi(x))+\frac{\partial F}{\partial y}(x, \phi(x)) \phi^{\prime}(x)=0, x \in J .$$

$$M(x, \phi(x))+N(x, \phi(x)) \phi^{\prime}(x)=0, x \in J,$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH53

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|Ordinary differential equations

The term ‘equatio differentialis’ (differential equations) was first used by Leibniz in 1676 to denote a relationship between the differentials of two variables. Very soon, this restricted usage was abandoned. Roughly speaking, differential equations are the equations involving one or more dependent variables (unknowns) and their derivatives/partial derivatives. If the unknown in the differential equation is a function of only one variable, then such differential equation is called an ordinary differential equation (ODE).
Notation: Unless specified otherwise, the unknown in the differential equation is denoted by $y$. Let $\mathbb{R}$ denote the set of real numbers, and $J$ be an open interval in $\mathbb{R}$. Throughout the book we denote the derivative of the function $y: J \rightarrow \mathbb{R}$ with respect to $x$ by either
$$\frac{d}{d x} y(x) \text { or } \frac{d y}{d x}(x) \text { or } y^{\prime}(x) .$$
When there is no ambiguity regarding the argument in the function $y$, we denote the derivative simply with $\frac{d y}{d x}$ or $y^{\prime}$. Similarly, let $y^{\prime \prime}$ and $y^{\prime \prime \prime}$ denote the second and the third derivative of $y$, respectively. In general, for $k \in \mathbb{N}$, $y^{(k)}$ or $\frac{d^k y}{d x^k}$ denotes the $k$-th order derivative of $y$.
With this notation, examples of ODEs are
$$\begin{gathered} \frac{d}{d x} y(x)=\left(\frac{d^2}{d x^2} y(x)\right)^5+y^2(x), x \in(0,1), \ y^{\prime}=3 y^2+(\sin x) y+\log \left(\cos ^2 y\right), x \in \mathbb{R} . \end{gathered}$$
The order of an ODE is the largest number $k$ such that the $k$-th order derivative of the unknown is present in the ODE. For example, the order of (1.1) is two.
At the beginning, it may look like tools from the integral calculus are sufficient to study ODEs. But very soon one realizes that to develop methods to solve or analyze them, one needs notions from subjects like analysis, linear algebra, etc. In fact, the study of differential equations motivated crucial development of many areas of mathematics: the theory of Fourier series and more general orthogonal expansions, integral transformations, Hilbert spaces, and Lebesgue integration to name a few.

## 数学代写|常微分方程代写ordinary differential equation代考|Applications of ODEs

Many laws in physics, chemistry, biology etc., can be easily expressed using differential equations. One of the reasons for this is the following. The quantity $y^{\prime}(x)$ can be interpreted as the rate of change of the quantity $y$ with respect to the quantity $x$. In many natural phenomena, there is a relationship between the unknowns (which are relatively difficult to measure), the rate of change of the unknowns with respect to a known quantity, and the other known quantities (which are easy to measure) that govern the process. When this relationship is expressed in mathematics, it turns out to be a (system of) differential equation(s). Therefore the study of ODEs is crucial in understanding physical sciences. In fact, much of the theory developed in ODEs owes to the questions/situations raised in the study of subjects like mechanics, astronomy, electronics etc.
Listing all the available ODE models in any branch of science is an impossible task. Therefore in this chapter, we present a few ODE models which arise from physics and biology which can be solved or analyzed using the material in the book. We begin with models from physics.

Example 1.2.1 (Radioactivity and half-life). Let $N(t)$ denote the number of radioactive active atoms in a substance of a fixed quantity at time $t$. Then a model for the decay of the number of radioactive atoms is
$$\begin{gathered} \frac{d}{d t} N(t)=-k N(t), t>0, \ N\left(t_0\right)=N_0, \end{gathered}$$
where $k>0$. Equation (1.3b) is known as the initial condition. This kind of models are studied in detail in Chapter 2, Subsection 2.1.3. One can easily verify that the solution to (1.3a) is
$$N(t)=N_0 e^{-k\left(t-t_0\right)}, t>t_0 .$$
The half-life of a specific radioactive isotope is defined as the time taken for half of its radioactive atoms to decay. In fact, the half-life is independent of the quantity of the radioactive material. We now calculate the half-life of an isotope using (1.3a) if $k$ is known explicitly. For, it is enough to find $T$ at which $N(T)=\frac{N_0}{2}$. From (1.4) we have
$$N(T)=N_0 e^{-k\left(T-t_0\right)}=\frac{N_0}{2}$$

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Ordinary differential equations

$$\frac{d}{d x} y(x) \text { or } \frac{d y}{d x}(x) \text { or } y^{\prime}(x) .$$

$$\frac{d}{d x} y(x)=\left(\frac{d^2}{d x^2} y(x)\right)^5+y^2(x), x \in(0,1), y^{\prime}=3 y^2+(\sin x) y+\log \left(\cos ^2 y\right), x \in \mathbb{R}$$
$\mathrm{ODE}$ 的阶数是最大数 $k$ 这样的 $k \mathrm{ODE}$ 中存在末知数的 -th 阶导数。例如，(1.1) 的阶数为二。

## 数学代写|常微分方程代写ordinary differential equation代考|Applications of ODEs

$$\frac{d}{d t} N(t)=-k N(t), t>0, N\left(t_0\right)=N_0,$$

$$N(t)=N_0 e^{-k\left(t-t_0\right)}, t>t_0 .$$

$$N(T)=N_0 e^{-k\left(T-t_0\right)}=\frac{N_0}{2}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|常微分方程代写ordinary differential equation代考|MATH2410

statistics-lab™ 为您的留学生涯保驾护航 在代写常微分方程ordinary differential equation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写常微分方程ordinary differential equation代写方面经验极为丰富，各种代写常微分方程ordinary differential equation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|常微分方程代写ordinary differential equation代考|Complex Solutions

One often encounters complex valued solutions of linear differential equations, and such solutions may be useful even though the equation itself has real valued coefficients and does not seem to invite a passage to the complex realm. For example, the equation $y^{(4)}+y=0$ has a solution basis comprising the four functions $e^{x / \sqrt{2}} \cos (x / \sqrt{2}), \quad e^{x / \sqrt{2}} \sin (x / \sqrt{2}), \quad e^{-x / \sqrt{2}} \cos (x / \sqrt{2}), \quad e^{-x / \sqrt{2}} \sin (x / \sqrt{2})$,

but if we allow complex valued solutions we can write another basis, algebraically simpler, comprising the functions
$$e^{\omega x}, \quad e^{i \omega x}, \quad e^{-\omega x}, \quad e^{-i \omega x},$$
where $\omega=(1+i) / \sqrt{2}$. The second basis might prove useful even when we seek, at the end of the day, a real valued solution. How these bases can be found is a topic taken later in this chapter.

For now we note that complex valued functions defined in the real interval $I$ form a vector space over the complex field $\mathbb{C}$. They can be differentiated in the obvious way, by differentiating the real part and the imaginary part, thus:
$$y^{\prime}(x)=u^{\prime}(x)+i v^{\prime}(x),$$
where $y(x)=u(x)+i v(x)$ and the functions $u$ and $v$ are real valued. It is now obvious that $u+i v$ is a solution of (1.5) (which has only real valued coefficient functions) if and only if $u$ and $v$ are individually real valued solutions. This says that the space of complex valued solutions is the complexification of the space of real solutions; it is a vector space over $\mathbb{C}$ with dimension $n$.

We can go further and suppose that the coefficient functions $p_1, \ldots, p_n$ have complex values, as well as the inhomogeneous term $g$ and the initial values. The analogues of the propositions of this section hold for complex equations without change, although they do not obviously follow from Proposition 1.3. The vector space of solutions of the homogeneous equation will be an $n$-dimensional vector space over $\mathbb{C}$ of complex valued functions. However, in this chapter we restrict ourselves to equations with real coefficients, as their properties can be derived from the as yet unproved Proposition 1.3.

It is important to understand that the independent variable $x$ is always real. At this point, we do not need differentiation with respect to a complex variable, which leads to the theory of complex analytic functions. The notion of a differential equation in the complex domain, for which a solution is a function of a complex variable, is not touched upon in this text.

## 数学代写|常微分方程代写ordinary differential equation代考|Homogeneous Linear Equations with Constant Coefficients

In this section we study the homogeneous equation
$$p_n y^{(n)}+p_{n-1} y^{(n-1)}+\cdots+p_1 y^{\prime}+p_0 y=0$$
with constant coefficients $p_0, \ldots, p_n$. We assume here that $p_n \neq 0$ so we could (but do not) divide throughout by $p_n$ to convert the equation to standard form. The solution space is an $n$-dimensional vector space of functions on the real line ]$-\infty, \infty[$. If the coefficients are real and we admit only real valued functions then it is $n$-dimensional over $\mathbb{R}$. If we admit complex valued solutions (and we are forced to do this if some coefficients are not real) then it is $n$-dimensional over $\mathbb{C}$. In this chapter we only study equations with real coefficients, but it may be still be advantageous to allow complex valued solutions.
Closely associated with the differential equation is the polynomial
$$P(X):=p_n X^n+p_{n-1} X^{n-1}+\cdots+p_1 X+p_0$$
and the so-called indicial equation ${ }^3$
$$P(X)=0 .$$
The roots of the indicial equation play a fundamental role in the theory of the linear equation with constant coefficients.
Proposition $1.11$

1. The function $e^{\lambda x}$ is a solution of (1.17) if and only if $\lambda$ is a root of the indicial equation.
2. Suppose that the indicial equation has $n$ distinct roots $\lambda_1, \ldots, \lambda_n$, possibly complex. Then the functions
$$e^{\lambda_1 x}, \ldots, e^{\lambda_n x}$$
form a solution basis.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Complex Solutions

$$e^{\omega x}, \quad e^{i \omega x}, e^{-\omega x}, \quad e^{-i \omega x},$$

$$y^{\prime}(x)=u^{\prime}(x)+i v^{\prime}(x)$$

## 数学代写|常微分方程代写ordinary differential equation代考|Homogeneous Linear Equations with Constant Coefficients

$$p_n y^{(n)}+p_{n-1} y^{(n-1)}+\cdots+p_1 y^{\prime}+p_0 y=0$$

$$P(X):=p_n X^n+p_{n-1} X^{n-1}+\cdots+p_1 X+p_0$$

$$P(X)=0 .$$

1. 功能 $e^{\lambda x}$ 是 (1.17) 的解当且仅当 $\lambda$ 是指示方程的根。
2. 假设指示方程有 $n$ 不同的根源 $\lambda_1, \ldots, \lambda_n$ ，可能很复杂。然后是函数
$$e^{\lambda_1 x}, \ldots, e^{\lambda_n x}$$
形成解决方案的基础。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。