## 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

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

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

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

## 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|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。

## 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|First order autonomous equations

Let us look at the simplest (nontrivial) case of a first-order autonomous equation and let us try to find the solution starting at a certain point $x_0$ at time $t=0$ :
$$\dot{x} f(x), \quad x(0) \quad x_0, \quad f \in C(\mathbb{R})$$

We could of course also ask for the solution starting at $x_0$ at time $t_0$. However, once we have a solution $\phi(t)$ with $\phi(0)=x_0$, the solution $\psi(t)$ with $\psi\left(t_0\right)=x_0$ is given by a simple shift $\psi(t)=\phi\left(t-t_0\right)$ (this holds in fact for any autonomous equation – compare Problem 1.8).

This equation can be solved using a small ruse. If $f\left(x_0\right) \neq 0$, we can divide both sides by $f(x)$ and integrate both sides with respect to $t$ :
$$\int_0^t \frac{\dot{x}(s) d s}{f(x(s))}=t \text {. }$$
Abbreviating $F(x)=\int_{x_0}^x \frac{d y}{f(y)}$ we see that every solution $x(t)$ of $(1.20)$ must satisfy $F(x(t))=t$. Since $F(x)$ is strictly monotone near $x_0$, it can be inverted and we obtain a unique solution
$$\phi(t)=F^{-1}(t), \quad \phi(0)=F^{-1}(0)=x_0,$$
of our initial value problem. Here $F^{-1}(t)$ is the inverse map of $F(t)$.
Now let us look at the maximal interval of existence. If $f\left(x_0\right)>0$ (the case $f\left(x_0\right)<0$ follows analogously), then $f$ remains positive in some interval $\left(x_1, x_2\right)$ around $x_0$ by continuity. Define $T_{+}=\lim {x \uparrow x_2} F(x) \in(0, \infty], \quad$ respectively $\quad T{-}=\lim {x \downarrow x_1} F(x) \in[-\infty, 0)$. (1.23) Then $\phi \in C^1\left(\left(T{-}, T_{+}\right)\right)$and $$\lim {t \uparrow T{+}} \phi(t)=x_2, \quad \text { respectively } \quad \lim {t \downarrow T{-}} \phi(t)=x_1 .$$ In particular, $\phi$ exists for all $t>0$ (resp. $t<0$ ) if and only if
$$T_{+}=\int_{x_0}^{x_2} \frac{d y}{f(y)}=+\infty,$$
that is, if $1 / f(x)$ is not integrable near $x_2$. Similarly, $\phi$ exists for all $t<0$ if and only if $1 / f(x)$ is not integrable near $x_1$.
Now let us look at some examples.

## 数学代写|常微分方程代写ordinary differential equation代考|Finding explicit solutions

We have seen in the previous section, that some differential equations can be solved explicitly. Unfortunately, there is no general recipe for solving a given differential equation. Moreover, finding explicit solutions is in general impossible unless the equation is of a particular form. In this section I will show you some classes of first-order equations which are explicitly solvable.
The general idea is to find a suitable change of variables which transforms the given equation into a solvable form. Hence we want to review this concept first. Given the point with coordinates $(t, x)$, we may change to new coordinates $(s, y)$ given by
$$s=\sigma(t, x), \quad y=\eta(t, x) .$$
Since we do not want to lose information, we require this transformation to be invertible.

A given function $\phi(t)$ will be transformed into a function $\psi(s)$ which has to be obtained by eliminating $t$ from
$$s=\sigma(t, \phi(t)), \quad \psi=\eta(t, \phi(t)) .$$
Unfortunately this will not always be possible (e.g., if we rotate the graph of a function in $\mathbb{R}^2$, the refiult might not be the graph of a function). To avoid this problem we restrict our attention to the special case of fiber preserving transformations
$$s=\sigma(t), \quad y=\eta(t, x)$$
(which map the fibers $t=$ const to the fibers $s=$ const). Denoting the inverse transform by
$$t=\tau(s), \quad x=\xi(s, y),$$
a straightforward application of the chain rule shows that $\phi(t)$ satisfies
$$\dot{x}=f(t, x)$$
if and only if $\psi(s)=\eta(\tau(s), \phi(\tau(s)))$ satisfies
$$\dot{y}=\dot{\tau}\left(\frac{\partial \eta}{\partial t}(\tau, \xi)+\frac{\partial \eta}{\partial x}(\tau, \xi) f(\tau, \xi)\right),$$
where $\tau=\tau(s)$ and $\xi=\xi(s, y)$.

## 数学代写|常微分方程代写ordinary differential equation代考|First order autonomous equations

$$\dot{x} f(x), \quad x(0) \quad x_0, \quad f \in C(\mathbb{R})$$

$$\int_0^t \frac{\dot{x}(s) d s}{f(x(s))}=t .$$

$$\phi(t)=F^{-1}(t), \quad \phi(0)=F^{-1}(0)=x_0,$$

$$T_{+}=\int_{x_0}^{x_2} \frac{d y}{f(y)}=+\infty$$

## 数学代写|常微分方程代写ordinary differential equation代考|Finding explicit solutions

$$s=\sigma(t, x), \quad y=\eta(t, x) .$$

$$s=\sigma(t, \phi(t)), \quad \psi=\eta(t, \phi(t)) .$$

$$s=\sigma(t), \quad y=\eta(t, x)$$
(映射纤维 $t=$ const 到纤维 $s=$ 常量)。将逆变换表示为
$$t=\tau(s), \quad x=\xi(s, y),$$

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

$$\dot{y}=\dot{\tau}\left(\frac{\partial \eta}{\partial t}(\tau, \xi)+\frac{\partial \eta}{\partial x}(\tau, \xi) f(\tau, \xi)\right),$$

## 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。

## 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|Newton’s equations

Let us begin with an example from physics. In classical mechanics a particle is described by a point in space whose location is given by a function

The derivative of this function with respect to time is the velocity of the particle
$$v=\dot{x}: \mathbb{R} \rightarrow \mathbb{R}^3$$
and the derivative of the velocity is the acceleration
$$a=\dot{v}: \mathbb{R} \rightarrow \mathbb{R}^3 .$$
In such a model the particle is usually moving in an external force field
$$F: \mathbb{R}^3 \rightarrow \mathbb{R}^3$$
which exerts a force $F(x)$ on the particle at $x$. The Newton’s second law states that, at each point $x$ in space, the force acting on the particle must be equal to the acceleration times the mass $m$ (a positive constant) of the particle, that is,
$$m \ddot{x}(t)=F(x(t)), \quad \text { for all } t \in \mathbb{R} .$$
Such a relation between a function $x(t)$ and its derivatives is called a differential equation. Equation (1.5) is of second order since the highest derivative is of second degree. More precisely, we have a system of differential equations since there is one for each coordinate direction.

In our case $x$ is called the dependent and $t$ is called the independent variable. It is also possible to increase the number of dependent variables by adding $v$ to the dependent variables and considering $(x, v) \in \mathbb{R}^6$. The advantage is, that we now have a first-order system
\begin{aligned} \dot{x}(t) &=v(t) \ \dot{v}(t) &=\frac{1}{m} F(x(t)) . \end{aligned}
This form is often better suited for theoretical investigations.
For given force $F$ one wants to find solutions, that is functions $x(t)$ that satisfy (1.5) (respectively (1.6)). To be more specific, let us look at the motion of a stone falling towards the earth. In the vicinity of the surface of the earth, the gravitational force acting on the stone is approximately constant and given by
$$F(x)=-m g\left(\begin{array}{l} 0 \ 0 \ 1 \end{array}\right) \text {. }$$

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

Let $U \subseteq \mathbb{R}^m, V \subseteq \mathbb{R}^n$ and $k \in \mathbb{N}_0$. Then $C^k(U, V)$ denotes the set of functions $U \rightarrow V$ having continuous derivatives up to order $k$. In addition, we will abbreviate $C(U, V)=C^0(U, V)$ and $C^k(U)=C^k(U, \mathbb{R})$.

A classical ordinary differential equation (ODE) is a relation of the form
$$F\left(t, x, x^{(1)}, \ldots, x^{(k)}\right)=0$$
for the unknown function $x \in C^k(J), J \subseteq \mathbb{R}$. Here $F \in C(U)$ with $U$ an open subset of $\mathbb{R}^{k+2}$ and
$$x^{(k)}(t)=\frac{d^k x(t)}{d t^k}, \quad k \in \mathbb{N}_0,$$
are the ordinary derivatives of $x$. One frequently calls $t$ the independent and $x$ the dependent variable. The highest derivative appearing in $F$ is called the order of the differential equation. A solution of the ODE (1.12) is a function $\phi \in C^k(I)$, where $I \subseteq J$ is an interval, such that
$$F\left(t, \phi(t), \phi^{(1)}(t), \ldots, \phi^{(k)}(t)\right)=0, \quad \text { for all } t \in I \text {. }$$
This implicitly implies $\left(t, \phi(t), \phi^{(1)}(t), \ldots, \phi^{(k)}(t)\right) \in U$ for all $t \in I$.
Unfortunately there is not too much one can say about general differential equations in the above form (1.12). Hence we will assume that one can solve $F$ for the highest derivative, resulting in a differential equation of the form
$$x^{(k)}=f\left(t, x, x^{(1)}, \ldots, x^{(k-1)}\right)$$
By the implicit function theorem this can be done at least locally near some point $(t, y) \in U$ if the partial derivative with respect to the highest derivative does not vanish at that point, $\frac{\partial F}{\partial y_k}(t, y) \neq 0$. This is the type of differential equations we will consider from now on.

## 数学代写|常微分方程代写ordinary differential equation代考|Newton’s equations

$$v=\dot{x}: \mathbb{R} \rightarrow \mathbb{R}^3$$

$$a=\dot{v}: \mathbb{R} \rightarrow \mathbb{R}^3 .$$

$$F: \mathbb{R}^3 \rightarrow \mathbb{R}^3$$

$$\dot{x}(t)=v(t) \dot{v}(t) \quad=\frac{1}{m} F(x(t)) .$$

$$F(x)=-m g\left(\begin{array}{llll} 0 & 0 & 1 \end{array}\right) .$$

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

$$F\left(t, x, x^{(1)}, \ldots, x^{(k)}\right)=0$$

$$x^{(k)}(t)=\frac{d^k x(t)}{d t^k}, \quad k \in \mathbb{N}_0,$$

$$F\left(t, \phi(t), \phi^{(1)}(t), \ldots, \phi^{(k)}(t)\right)=0, \quad \text { for all } t \in I .$$

$$x^{(k)}=f\left(t, x, x^{(1)}, \ldots, x^{(k-1)}\right)$$

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## MATLAB代写

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