## 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。

## 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|The Wronskian

Let $u_1, \ldots, u_n$ be a sequence of $n$ functions each of which is $n-1$ times differentiable in an interval $I$. We do not assume that they are the solutions of any differential equation. The determinant
$$W\left(u_1, \ldots, u_n\right)(x):=\left|\begin{array}{ccc} u_1(x) & \ldots & u_n(x) \ u_1^{\prime}(x) & \ldots & u_n^{\prime}(x) \ \vdots & & \vdots \ u_1^{(n-1)}(x) & \ldots & u_n^{(n-1)}(x) \end{array}\right|$$
is called the Wronskian of the functions $u_1, \ldots, u_n$. It is a function on the interval $I$.
Proposition 1.5 Let the coefficient functions $p_0, \ldots, p_{n-1}$ be continuous in the open interval I. Let $u_1, \ldots, u_n$ be $n$ solutions of (1.5) on the interval I and let $x_0 \in I$. Then the solutions $u_1, \ldots, u_n$ form a basis for the space $E$ of all solutions of $(1.5)$ on I if and only if $W\left(u_1, \ldots, u_n\right)\left(x_0\right) \neq 0$.

Proof In the proof of Proposition $1.4$ we saw that the mapping $\kappa: E \rightarrow \mathbb{R}^n$, that maps each solution to its vector of Cauchy data at $x_0$, is a vector space isomorphism. It follows that $n$ solutions $u_1, \ldots, u_n$ form a basis of $E$ if and only if the $n$ vectors $\kappa\left(u_1\right), \ldots, \kappa\left(u_n\right)$ form a basis of $\mathbb{R}^n$. The necessary and sufficient condition for this is that the determinant $W\left(u_1, \ldots, u_n\right)\left(x_0\right)$ is not 0 , since these vectors constitute its columns.

Observing that the point $x_0$ in Proposition $1.5$ can be any point in the interval $I$, we obtain the following:

Proposition 1.6 Let the coefficient functions $p_0, \ldots, p_{n-1}$ be continuous in the open interval I. Let $u_1, \ldots, u_n$ be $n$ solutions of (1.5) on the interval I. Then either the Wronskian $W\left(u_1, \ldots, u_n\right)(x)$ is non-zero for every $x$ in I or it is zero for every $x$ in $I$.

Even though Proposition $1.6$ follows at once from Proposition $1.5$ there is an interesting formula due to Abel that leads to the same conclusion. It shows that $W\left(u_1, \ldots, u_n\right)(x)$ satisfies a first order linear differential equation.

## 数学代写|常微分方程代写ordinary differential equation代考|Non-homogeneous Equations

Proposition $1.8$ Let the coefficient functions $p_0, \ldots, p_{n-1}$ be continuous in the open interval I and let the function $g$ be continuous in I. Let $v(x)$ be a solution on I of the non-homogeneous equation
$$y^{(n)}+p_{n-1}(x) y^{(n-1)}+\cdots+p_1(x) y^{\prime}+p_0(x) y=g(x) .$$

Let $u_1, \ldots, u_n$ be a solution basis for the homogeneous equation. Then the general solution of (1.8) can be written
$$y(x)=c_1 u_1(x)+\cdots+c_n u_n(x)+v(x)$$
where $c_1, \ldots, c_n$ are arbitrary constants.
The function $v(x)$ in this proposition is traditionally called a particular solution of the non-homogeneous equation. In spite of its name, any solution can be used as a particular solution.

Proof In the first place the formula (1.9) is a solution of (1.8) however the constants $c_1, \ldots, c_n$ are chosen. Now let $y(x)$ be a solution of (1.8). Consider the function $z(x):=y(x)-v(x)$. It is easily seen that $z(x)$ satisfies the homogeneous equation, and hence is of the form $c_1 u_1(x)+\cdots+c_n u_n(x)$ for a certain choice of the constants.

Next we prove the existence of a particular solution. In fact we do more. We show that once a solution basis is known for the homogeneous equation a particular solution can be found by quadratures. The classical procedure is called the method of variation of parameters.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|Non-homogeneous Equations

$$y^{(n)}+p_{n-1}(x) y^{(n-1)}+\cdots+p_1(x) y^{\prime}+p_0(x) y=g(x) .$$

$$y(x)=c_1 u_1(x)+\cdots+c_n u_n(x)+v(x)$$

## 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|First Order Linear Equations

The first order linear equation is
$$y^{\prime}+p(x) y=g(x)$$
By a general solution of a differential equation is meant a function of $x$ involving arbitrary constants $A, B, \ldots$ which is a solution for all values of the constants and is such that all possible solutions are obtained by assigning appropriate values to the constants.

Obviously a general solution is not unique (just as a vector space does not have a unique basis). Nevertheless, we often use the definite article and speak of “the general solution”, somehow regarding them as being equivalent to each other. Sometimes a general solution is referred to as a complete solution.

Proposition 1.1 If $p$ and $g$ are continuous in the interval 1 then a general solution of (1.3) is given by
$$y(x)=e^{-M(x)}\left(C+\int g(x) e^{M(x)} d x\right)$$
where $C$ is an arbitrary constant, $M(x):=\int p(x) d x$ is an antiderivative of $p(x)$ in I and $\int g(x) e^{M(x)} d x$ is an antiderivative of $g(x) e^{M(x)}$ in I.
Notes
(1) A formula such as (1.4) that solves a differential equation by means of antiderivatives is traditionally called a solution by quadrature, that being an old name for integration. The first order linear equation requires in general two quadratures.
(2) A function $f$ continuous in an interval $I$ possesses an antiderivative in $I$. It is a solution of the simplest possible differential equation $d y / d x=f(x)$. The fundamental theorem of calculus provides one of the form
$$F(x)=\int_{x_0}^x f(t) d t, \quad(x \in I)$$
where $x_0$ is an arbitrarily chosen base point in $I$ (but there may be others not obtainable by selecting a base point $x_0$ ).
(3) If $F$ is an antiderivative of $f$ then so is $F+C$ where $C$ is any constant function. Any two antiderivatives of $F$ in the same interval differ by a constant.

## 数学代写|常微分方程代写ordinary differential equation代考|The nth Order Linear Equation

No method is known that solves the general $n$th order, linear differential equation by quadratures if $n \geq 2$. Only special cases, such as equations with constant coefficients, can be solved by these methods.
Functions on an interval $I$ can be added pointwise to build new functions
$$\left(f_1+f_2\right)(x)=f_1(x)+f_2(x), \quad(x \in I) .$$
They can be multiplied by scalars
$$(\lambda f)(x)=\lambda f(x), \quad(x \in I) .$$
They therefore form a vector space over the real number field $\mathbb{R}$. All references to vector spaces, for the time heing, will mean vector spaces over the field $R$.
The following proposition is now easy to check (and the reader should do it):
Proposition 1.2 The solutions of an nth order, homogeneous, linear differential equation
$$y^{(n)}+p_{n-1}(x) y^{(n-1)}+\cdots+p_1(x) y^{\prime}+p_0(x) y=0$$
constitute a vector space of functions on the interval $I$.

Of course the zero function $y=0$ is a solution of (1.5) but $a$ priori we do not know whether there are others if $n \geq 2$ (the case $n=1$ being satisfactorily dealt with by Proposition 1.1). We now state the fundamental existence theorem for (1.5). We shall prove it in a later chapter.

Proposition $1.3$ Let the coefficient functions $p_0, \ldots, p_{n-1}$ be continuous in the open interval I. Let $x_0 \in I$ and let numbers $a_1, \ldots, a_n$ be given. Then the problem (1.5) has a unique solution in the interval I that satisfies the $n$ conditions
$$y\left(x_0\right)=a_1, \quad y^{\prime}\left(x_1\right)=a_2, \quad \ldots \quad y^{(n-1)}\left(x_0\right)=a_n$$
Problem (1.5) together with the conditions (1.6) is called the initial value problem, or Cauchy problem, for the differential equation (1.5). The conditions (1.6) are called the initial conditions or Cauchy conditions. From Proposition $1.3$ we now deduce a remarkable result:

Proposition 1.4 Let the coefficient functions $p_0, \ldots, p_{n-1}$ be continuous in the open interval 1 . Then the space of solutions of (1.5) is an n-dimensional vector space of functions defined in the interval $I$.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|First Order Linear Equations

$$y^{\prime}+p(x) y=g(x)$$

$$y(x)=e^{-M(x)}\left(C+\int g(x) e^{M(x)} d x\right)$$

Notes
(1) 诸如 (1.4) 之类的通过反导数求解微分方程的公式传统上称为求积法解，这是积分的旧名称。一阶线性 方程通常需要两个正交。
(2) 一个函数 $f$ 在一个区间内连续 $I$ 具有反导数 $I$. 它是最简单的微分方程的解 $d y / d x=f(x)$. 微积分的基 本定理提供了一种形式
$$F(x)=\int_{x_0}^x f(t) d t, \quad(x \in I)$$

(3) 如果 $F$ 是的反导数 $f$ 那么也是 $F+C$ 在哪里 $C$ 是任何常数函数。的任意两个反导数 $F$ 在同一区间内相差 一个常数。

## 数学代写|常微分方程代写ordinary differential equation代考|The nth Order Linear Equation

$$\left(f_1+f_2\right)(x)=f_1(x)+f_2(x), \quad(x \in I) .$$

$$(\lambda f)(x)=\lambda f(x), \quad(x \in I) .$$

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

$$y\left(x_0\right)=a_1, \quad y^{\prime}\left(x_1\right)=a_2, \quad \ldots \quad y^{(n-1)}\left(x_0\right)=a_n$$

## 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|MATH289

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|Reparametrization of Time

Suppose that $U$ is an open set in $\mathbb{R}^n, f: U \rightarrow \mathbb{R}^n$ is a smooth function, and $g: U \rightarrow \mathbb{R}$ is a positive smooth function. What is the relationship among the solutions of the differential equations
\begin{aligned} \dot{x} & =f(x), \ \dot{x} & =g(x) f(x) ? \end{aligned}
The vector fields defined by $f$ and $g f$ have the same direction at each point in $U$, only their lengths are different. Thus, by our geometric interpretation of autonomous differential equations, it is intuitively clear that the differential equations (1.10) and (1.11) have the same phase portraits in $U$. This fact is a corollary of the next proposition.

Proposition 1.14. If $J \subset \mathbb{R}$ is an open interval containing the origin and $\gamma: J \rightarrow \mathbb{R}^n$ is a solution of the differential equation (1.10) with $\gamma(0)=$ $x_0 \in U$, then the function $B: J \rightarrow \mathbb{R}$ given by
$$B(t)=\int_0^t \frac{1}{g(\gamma(s))} d s$$
is invertible on its range $K \subseteq \mathbb{R}$. If $\rho: K \rightarrow J$ is the inverse of $B$, then the identity
$$\rho^{\prime}(t)=g(\gamma(\rho(t))$$
holds for all $t \in K$, and the function $\sigma: K \rightarrow \mathbb{R}^n$ given by $\sigma(t)=\gamma(\rho(t))$ is the solution of the differential equation (1.11) with initial condition $\sigma(0)=$ $x_0$

## 数学代写|常微分方程代写ordinary differential equation代考|Stability and the Direct Method of Lyapunov

Let us consider a rest point $x_0$ for the autonomous differential equation
$$\dot{x}=f(x), \quad x \in \mathbb{R}^n .$$
A continuous function $V: U \rightarrow \mathbb{R}$, where $U \subseteq \mathbb{R}^n$ is an open set with $x_0 \in U$, is called a Lyapunov function for the differential equation (1.15) at $x_0$ provided that

(i) $V\left(x_0\right)=0$,
(ii) $V(x)>0$ for $x \in U-\left{x_0\right}$,
(iii) the function $x \mapsto \operatorname{grad} V(x)$ is continuous for $x \in U-\left{x_0\right}$, and, on this set, $\dot{V}(x):=\operatorname{grad} V(x) \cdot f(x) \leq 0$.
(iv) $\dot{V}(x)<0$ for $x \in U-\left{x_0\right}$
then $V$ is called a strict Lyapunov function.
Theorem $1.30$ (Lyapunov’s Stability Theorem). If $x_0$ is a rest point for the differential equation (1.15) and $V$ is a Lyapunov function for the system al $x_0$, then $x_0$ is slable. If, in addilion, $V$ is a slricl Lyapunov function, then $x_0$ is asymptotically stable.

The idea of Lyapunov’s method is very simple. In many cases the level sets of $V$ are “spheres” surrounding the rest point $x_0$ as in Figure 1.10. Suppose this is the case and let $\phi_t$ denote the flow of the differential equation (1.15). If $y$ is in the level set $\mathcal{S}c=\left{x \in \mathbb{R}^n: V(x)=c\right}$ of the function $V$, then, by the chain rule, we have that $$\left.\frac{d}{d t} V\left(\phi_t(y)\right)\right|{t=0}=\operatorname{grad} V(y) \cdot f(y) \leq 0$$

## 数学代写|常微分方程代写ordinary differential equation代考|Reparametrization of Time

$$\dot{x}=f(x), \dot{x} \quad=g(x) f(x) ?$$

$$B(t)=\int_0^t \frac{1}{g(\gamma(s))} d s$$

$$\rho^{\prime}(t)=g(\gamma(\rho(t))$$

## 数学代写|常微分方程代写ordinary differential equation代考|Stability and the Direct Method of Lyapunov

$$\dot{x}=f(x), \quad x \in \mathbb{R}^n .$$

(一世) $V\left(x_0\right)=0$ ，
(ii) $V(x)>0$ 为了 $\mathrm{x} \backslash \mathrm{in} \mathrm{U}$-Veft{{_O\right } } \text { , }
(iii) 函数 $x \mapsto \operatorname{grad} V(x)$ 是连续的 $\mathrm{x} \backslash$ in U-\left{x_0\right } } \text { ，并且，在这个集合上， } $\dot{V}(x):=\operatorname{grad} V(x) \cdot f(x) \leq 0$

(iv) $\dot{V}(x)<0$ 为了 $\mathrm{x} \backslash$ in U-\left } { \mathrm { x } _ { – } \text { 아ight } }

Lyapunov 方法的思想非常简单。在许多情况下，水平集 $V$ 是围绕休息点的“球体” $x_0$ 如图 $1.10$ 所示。假设 是这种情况，让 $\phi_t$ 表示微分方程 (1.15) 的流向。如果 $y$ 在水平集中
$$\frac{d}{d t} V\left(\phi_t(y)\right) \mid t=0=\operatorname{grad} V(y) \cdot f(y) \leq 0$$

## 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|Geometric Interpretation of Autonomous Systems

In this section we will describe a very important geometric interpretation of the autonomous differential equation
$$\dot{x}=f(x), \quad x \in \mathbb{R}^n .$$
The function given by $x \mapsto(x, f(x))$ defines a vector field on $\mathbb{R}^n$ associated with the differential equation (1.7). Here the first component of the function specifies the base point and the second component specifies the vector at this base point. A solution $t \mapsto \phi(t)$ of (1.7) has the property that its tangent vector at each time $t$ is given by
$$(\phi(t), \dot{\phi}(t))=(\phi(t), f(\phi(t))) .$$
In other words, if $\xi \in \mathbb{R}^n$ is on the orbit of this solution, then the tangent line to the orbit at $\xi$ is generated by the vector $(\xi, f(\xi))$, as depicted in Figure 1.1.

We have just mentioned two essential facts: $(i)$ There is a one-to-one correspondence between vector fields and autonomous differential equations. (ii) Every tangent vector to a solution curve is given by a vector in the vector field. These facts suggest that the geometry of the associated vector field is closely related to the geometry of the solutions of the differential equation when the solutions are viewed as curves in a Euclidean space. This geometric interpretation of the solutions of autonomous differential equations provides a deep insight into the general nature of the solutions of differential equations, and at the same time suggests the “geometric method” for studying differential equations: qualitative features expressed geometrically are paramount; analytic formulas for solutions are of secondary importance. Finally, let us note that the vector field associated with a differential equation is given explicitly. Thus, one of the main goals of the geometric method is to derive qualitative properties of solutions directly from the vector field without “solving” the differential equation.

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

An important property of the set of solutions of the autonomous differential equation (1.7),
$$\dot{x}=f(x), \quad x \in \mathbb{R}^n,$$
is the fact that these solutions form a one-parameter group that defines a phase flow. More precisely, let us define the function $\phi: \mathbb{R} \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ as follows: For $x \in \mathbb{R}^n$, let $t \mapsto \phi(t, x)$ denote the solution of the autonomous differential equation (1.7) such that $\phi(0, x)=x$.

We know that solutions of a differential equation may not exist for all $t \in \mathbb{R}$. However, for simplicity, let us assume that every solution does exist for all time. If this is the case, then each solution is called complete, and the fact that $\phi$ defines a one-parameter group is expressed concisely as follows:
$$\phi(t+s, x)=\phi(t, \phi(s, x)) .$$
In view of this equation, if the solution starting at time zero at the point $x$ is continued until time $s$, when it reaches the point $\phi(s, x)$, and if a new solution at this point with initial time zero is continued until time $t$, then this new solution will reach the same point that would have been reached if the original solution, which started at time zero at the point $x$, is continued until time $t+s$.

The prototypical example of a flow is provided by the general solution of the ordinary differential equation $\dot{x}=a x, x \in \mathbb{R}, a \in \mathbb{R}$. The solution is given by $\phi\left(t, x_0\right)=e^{a t} x_0$, and it satisfies the group property
$$\phi\left(t+s, x_0\right)=e^{a(t+s)} x_0=e^{a t}\left(e^{a s} x_0\right)=\phi\left(t, e^{a s} x_0\right)=\phi\left(t, \phi\left(s, x_0\right)\right) .$$
For the general case, let us suppose that $t \mapsto \phi(t, x)$ is the solution of the differential equation (1.7). Fix $s \in \mathbb{R}, x \in \mathbb{R}^n$, and define
$$\psi(t):=\phi(t+s, x), \quad \gamma(t):=\phi(t, \phi(s, x)) .$$

## 数学代写|常微分方程代写ordinary differential equation代考|Geometric Interpretation of Autonomous Systems

$$\dot{x}=f(x), \quad x \in \mathbb{R}^n .$$

$$(\phi(t), \dot{\phi}(t))=(\phi(t), f(\phi(t))) .$$

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

$$\dot{x}=f(x), \quad x \in \mathbb{R}^n,$$

$$\phi(t+s, x)=\phi(t, \phi(s, x)) .$$

$$\phi\left(t+s, x_0\right)=e^{a(t+s)} x_0=e^{a t}\left(e^{a s} x_0\right)=\phi\left(t, e^{a s} x_0\right)=\phi\left(t, \phi\left(s, x_0\right)\right) .$$

$$\psi(t):=\phi(t+s, x), \quad \gamma(t):=\phi(t, \phi(s, x))$$

## 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|Existence and Uniqueness

Let $J \subseteq \mathbb{R}, U \subseteq \mathbb{R}^n$, and $\Lambda \subseteq \mathbb{R}^k$ be open subsets, and suppose that $f: J \times U \times \Lambda \rightarrow \mathbb{R}^n$ is a smooth function. Here the term “smooth” means that the function $f$ is continuously differentiable. An ordinary differential equation (ODE) is an equation of the form
$$\dot{x}=f(t, x, \lambda)$$
where the dot denotes differentiation with respect to the independent variable $t$ (usually a measure of time), the dependent variable $x$ is a vector of state variables, and $\lambda$ is a vector of parameters. As convenient terminology, especially when we are concerned with the components of a vector differential equation, we will say that equation (1.1) is a system of differential equations. Also, if we are interested in changes with respect to parameters, then the differential equation is called a family of differential equations.
Example 1.1. The forced van der Pol oscillator
\begin{aligned} & \dot{x}_1=x_2, \ & \dot{x}_2=b\left(1-x_1^2\right) x_2-\omega^2 x_1+a \cos \Omega t \end{aligned}
is a differential equation with $J=\mathbb{R}, x=\left(x_1, x_2\right) \in U=\mathbb{R}^2$,
$$\Lambda=\left{(a, b, \omega, \Omega):(a, b) \in \mathbb{R}^2, \omega>0, \Omega>0\right},$$
and $f: \mathbb{R} \times \mathbb{R}^2 \times \Lambda \rightarrow \mathbb{R}^2$ defined in components by
$$\left(t, x_1, x_2, a, b, \omega, \Omega\right) \mapsto\left(x_2, b\left(1-x_1^2\right) x_2-\omega^2 x_1+a \cos \Omega t\right) .$$
If $\lambda \in \Lambda$ is fixed, then a solution of the differential equation (1.1) is a function $\phi: J_0 \rightarrow U$ given by $t \mapsto \phi(t)$, where $J_0$ is an open subset of $J$, such that
$$\frac{d \phi}{d t}(t)=f(t, \phi(t), \lambda)$$
for all $t \in J_0$.
In this context, the words “trajectory,” “phase curve,” and “integral curve” are also used to refer to solutions of the differential equation (1.1). However, it is useful to have a term that refers to the image of the solution in $\mathbb{R}^n$. Thus, we define the orbit of the solution $\phi$ to be the set ${\phi(t) \in U$ : $\left.t \in J_0\right}$

## 数学代写|常微分方程代写ordinary differential equation代考|Types of Differential Equations

Differential equations may be classified in several different ways. In this section we note that the independent variable may be implicit or explicit, and that higher order derivatives may appear.
An autonomous differential equation is given by
$$\dot{x}=f(x, \lambda), \quad x \in \mathbb{R}^n, \quad \lambda \in \mathbb{R}^k ;$$
that is, the function $f$ does not depend explicitly on the independent variable. If the function $f$ does depend explicitly on $t$, then the corresponding differential equation is called nonautonomous.

In physical applications, we often encounter equations containing second, third, or higher order derivatives with respect to the independent variable. These are called second order differential equations, third order differential equations, and so on, where the the order of the equation refers to the order of the highest order derivative with respect to the independent variable that appears explicitly. In this hierarchy, a differential equation is called a first order differential equation.

Recall that Newton’s second law-the rate of change of the linear momentum acting on a body is equal to the sum of the forces acting on the body -involves the second derivative of the position of the body with respect to time. Thus, in many physical applications the most common differential equations used as mathematical models are second order differential equations. For example, the natural physical derivation of van der Pol’s equation leads to a second order differential equation of the form
$$\ddot{u}+b\left(u^2-1\right) \dot{u}+\omega^2 u=a \cos \Omega t .$$
An essential fact is that every differential equation is equivalent to a first order system. To illustrate, let us consider the conversion of van der Pol’s equation to a first order system. For this, we simply define a new variable $v:=\dot{u}$ so that we obtain the following system:
\begin{aligned} \dot{u} & =v \ \dot{v} & =-\omega^2 u+b\left(1-u^2\right) v+a \cos \Omega t . \end{aligned}

## 数学代写|常微分方程代写ordinary differential equation代考|Existence and Uniqueness

$$\dot{x}=f(t, x, \lambda)$$

$$\dot{x}_1=x_2, \quad \dot{x}_2=b\left(1-x_1^2\right) x_2-\omega^2 x_1+a \cos \Omega t$$

ILambda $=\backslash$ eft $\left{\left(a, b\right.\right.$, Iomega, \Omega):(a, b) \in $\backslash m a t h b b{R}^{\wedge} 2$, lomega $>0$, \Omega $>0$ \right } } \text { , }

$$\left(t, x_1, x_2, a, b, \omega, \Omega\right) \mapsto\left(x_2, b\left(1-x_1^2\right) x_2-\omega^2 x_1+a \cos \Omega t\right) .$$

$$\frac{d \phi}{d t}(t)=f(t, \phi(t), \lambda)$$

## 数学代写|常微分方程代写ordinary differential equation代考|Types of Differential Equations

$$\dot{x}=f(x, \lambda), \quad x \in \mathbb{R}^n, \quad \lambda \in \mathbb{R}^k ;$$

$$\ddot{u}+b\left(u^2-1\right) \dot{u}+\omega^2 u=a \cos \Omega t .$$

$$\dot{u}=v \dot{v} \quad=-\omega^2 u+b\left(1-u^2\right) v+a \cos \Omega t .$$

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## 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

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• 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|Qualitative analysis of first-order equations

As already noted in the previous section, only very few ordinary differential equations are explicitly solvable. Fortunately, in many situations a solution is not needed and only some qualitative aspects of the solutions are of interest. For example, does it stay within a certain region, what does it look like for large $t$, etc.

Moreover, even in situations where an exact solution can be obtained, a qualitative analysis can give a better overview of the behavior than the formula for the solution. For example, consider the logistic growth model (Problem 1.16)
$$\dot{x}(t)=(1-x(t)) x(t)-h,$$
which can be solved by separation of variables. To get an overview we plot the corresponding right hand side $f(x)=(1-x) x-h$ : Since the sign of $f(x)$ tells us in what direction the solution will move, all we have to do is to discuss the sign of $f(x)$ ! For $0<h<\frac{1}{4}$ there are two zeros $x_{1,2}=\frac{1}{2}(1 \pm \sqrt{1-4 h})$. If we start at one of these zeros, the solution will stay there for all $t$. If we start below $x_1$ the solution will decrease and converge to $-\infty$. If we start above $x_1$ the solution will increase and converge to $x_2$. If we start above $x_2$ the solution will decrease and again converge to $x_2$.

At $h=\frac{1}{4}$ a bifurcation occurs: The two zeros coincide $x_1=x_2$ but otherwise the analysis from above still applies. For $h>\frac{1}{4}$ there are no zeros and all solutions decrease and converge to $-\infty$.

So we get a complete picture just by discussing the sign of $f(x)$ ! More generally we have the following result for the first-order autonomous initial value problem (Problem 1.27)
$$\dot{x}=f(x), \quad x(0)=x_0,$$
where $f$ is such that solutions are unique (e.g. $f \in C^1$ ).
(i) If $f\left(x_0\right)=0$, then $x(t)=x_0$ for all $t$.
(ii) If $f\left(x_0\right) \neq 0$, then $x(t)$ converges to the first zero left $\left(f\left(x_0\right)<0\right)$ respectively right $\left(f\left(x_0\right)>0\right)$ of $x_0$. If there is no such zero the solution converges to $-\infty$, respectively $\infty$.

If our differential equation is not autonomous, the situation becomes a bit more involved. As a prototypical example let us investigate the differential equation
$$\dot{x}=x^2-t^2 .$$
It is of Riccati type and according to the previous section, it cannot be solved unless a particular solution can be found. But there does not seem to be a solution which can be easily guessed. (We will show later, in Problem 4.8, that it is explicitly solvable in terms of special functions.)

## 数学代写|常微分方程代写ordinary differential equation代考|Qualitative analysis of first-order periodic equations

Some of the most interesting examples are periodic ones, where $f(t+1, x)=$ $f(t, x)$ (without loss we have assumed the period to be one). So let us consider the logistic growth model with a time dependent harvesting term
$$\dot{x}(t)=(1-x(t)) x(t)-h \cdot(1-\sin (2 \pi t)),$$
where $h \geq 0$ is some positive constant. In fact, we could replace $1-\sin (2 \pi t)$ by any nonnegative periodic function $g(t)$ and the analysis below will still hold.

The solutions corresponding to some initial conditions for $h=0.2$ are depicted below.

It looks like all solutions starting above some value $x_1$ converge to a periodic solution starting at some other value $x_2>x_1$, while solutions starting below $x_1$ diverge to $-\infty$.

They key idea is to look at the fate of an arbitrary initial value $x$ after precisely one period. More precisely, let us denote the solution which starts at the point $x$ at time $t=0$ by $\phi(t, x)$. Then we can introduce the Poincaré map via
$$P(x)=\phi(1, x) .$$
By construction, an initial condition $x_0$ will correspond to a periodic solution if and only if $x_0$ is a fixed point of the Poincaré map, $P\left(x_0\right)=x_0$. In fact, this follows from uniqueness of solutions of the initial value problem, since $\phi(t+1, x)$ again satisfies $\dot{x}=f(t, x)$ if $f(t+1, x)=f(t, x)$. So $\phi\left(t+1, x_0\right)=\phi\left(t, x_0\right)$ if and only if equality holds at the initial time $t=0$, that is, $\phi\left(1, x_0\right)=\phi\left(0, x_0\right)=x_0$.

We begin by trying to compute the derivative of $P(x)$ as follows. Set
$$\theta(t, x)=\frac{\partial}{\partial x} \phi(t, x)$$
and differentiate the differential equation
$$\dot{\phi}(t, x)=(1-\phi(t, x)) \phi(t, x)-h \cdot(1-\sin (2 \pi t)),$$
with respect to $x$ (we will justify this step in Theorem 2.10). Then we obtain
$$\dot{\theta}(t, x)=(1-2 \phi(t, x)) \theta(t, x)$$
and assuming $\phi(t, x)$ is known we can use Problem $1.13$ to write down the solution
$$\theta(t, x)=\exp \left(\int_0^t(1-2 \phi(s, x)) d s\right) .$$

## 数学代写|常微分方程代写ordinary differential equation代考|Qualitative analysis of first-order equations

$$\dot{x}(t)=(1-x(t)) x(t)-h,$$

$$\dot{x}=f(x), \quad x(0)=x_0,$$

$(一)$ 如果 $f\left(x_0\right)=0$ ，然后 $x(t)=x_0$ 对所有人 $t$.
(ii) 如果 $f\left(x_0\right) \neq 0$ ，然后 $x(t)$ 收玫到左边的第一个零 $\left(f\left(x_0\right)<0\right)$ 分别对 $\left(f\left(x_0\right)>0\right)$ 的 $x_0$. 如果没有这样的 零，则解收敛到 $-\infty$ ，分别 $\infty$.

$$\dot{x}=x^2-t^2 .$$

## 数学代写|常微分方程代写ordinary differential equation代考|Qualitative analysis of first-order periodic equations

$$\dot{x}(t)=(1-x(t)) x(t)-h \cdot(1-\sin (2 \pi t)),$$

$$P(x)=\phi(1, x) .$$

$$\theta(t, x)=\frac{\partial}{\partial x} \phi(t, x)$$

$$\dot{\phi}(t, x)=(1-\phi(t, x)) \phi(t, x)-h \cdot(1-\sin (2 \pi t)),$$

$$\dot{\theta}(t, x)=(1-2 \phi(t, x)) \theta(t, x)$$

$$\theta(t, x)=\exp \left(\int_0^t(1-2 \phi(s, x)) d s\right) .$$

## 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。