## 数学代写|常微分方程代写ordinary differential equation代考|MAT-328

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。

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

The general form of such equations is, according to the introduction (see e.g.(15))
$$a_0(x) y^{\prime \prime}+a_1(x) y^{\prime}+a_2(x) y=b(x),$$
where $a_0, a_1, a_2, b$ are real functions defined on a real interval $I \subseteq \Re$. We may consider these functions continuous on $I$.
If $a_0(x) \neq 0, \forall x \in I$, we can divide both members of (1.2.1) by it, thus getting an equation whose leading coefficient is 1

$$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=f(x)$$
where we used the notations $p(x)=\frac{a_1(x)}{a_0(x)}, q(x)=\frac{a_2(x)}{a_0(x)}, f(x)=\frac{b(x)}{a_0(x)}$. Obviously, if the coefficients of (1.2.1) are of class $\mathrm{C}^0(I)$, so are $p, q$ and $f$.
We see that, if $a_0(x)=0, \forall x \in I$, the equation is no more of second order, and, at the points at which $a_0(x)=0$, it has singularities. For the moment, we shall not deal with such situations, such that we consider that the given equation may be brought to the form (1.2.2).
Let us denote by
$$\mathrm{L} y \equiv y^{\prime \prime}+p(x) y^{\prime}+q(x) y .$$
The operator $\mathrm{L}$ is defined on $\mathrm{C}^2(I)$, with range in $\mathrm{C}^0(I)$, and we can easily prove that it is linear.
The kernel of this operator is a subset of $\mathrm{C}^2(I)$, containing functions cancelled by $\mathrm{L}$
$$\operatorname{ker} \mathrm{L}=\left{y \in \mathrm{C}^2(I) \mid \mathrm{L} y=0\right} .$$

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

Let us take the associated to (1.2.1) homogeneous equation
$$a_0(x) y^{\prime \prime}+a_1(x) y^{\prime}+a_2(x) y=0 .$$
If we know a particular solution of this equation, say $Y(x)$, we can completely solve (1.2.7). Indeed, let us perform the change of function
$$y(x)=z(x) Y(x),$$
$z(x)$ being the new unknown function. Replacing this in (1.2.7), we get

$$a_0(x) Y z^{\prime \prime}+\left[2 a_0(x) Y^{\prime}+a_1 Y\right] z^{\prime}+\left[a_0(x) Y^{\prime \prime}+a_1(x) Y^{\prime}+a_2(x) Y\right] z=0 .$$
As $Y$ is a solution of (1.2.7), it follows that $u=z^{\prime}$ must satisfy
$$a_0(x) Y u^{\prime}+\left[2 a_0(x) Y^{\prime}+a_1 Y\right] u=0 ;$$
this is a linear first order ODE.
We conclude that if we know a particular solution, we can reduce the order of the given equation by one unit.
Suppose now that $Y_1(x)$ is a known particular solution of the homogeneous equation, associated to (1.2.2)
$$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$
and suppose moreover that $Y_1$ does not vanish on $I$. Using the change of function $y=Y_1 z$, we find that $u=z^{\prime}$ must satisfy
$$u^{\prime}+\left(2 \frac{Y_1^{\prime}(x)}{Y_1(x)}+p(x)\right) u=0,$$
i.e., a linear first order homogeneous ordinary differential equation. According to Sec.1.2, it allows the general integral
$$u(x)=C_1 \frac{e^{-\int p(x) \mathrm{d} x}}{Y_1^2(x)},$$
where $\int p(x) \mathrm{d} x$ is a primitive of $p(x)$ and $C_1$ is an arbitrary constant. Getting back to $y$, we deduce
$$y(x)=C_1 Y_1(x) \int \frac{e^{-\int p(x) \mathrm{d} x}}{Y_1^2(x)} \mathrm{d} x .$$

# 常微分方程代写

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

$$a_0(x) y^{\prime \prime}+a_1(x) y^{\prime}+a_2(x) y=b(x),$$

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

$$\mathrm{L} y \equiv y^{\prime \prime}+p(x) y^{\prime}+q(x) y .$$

$$\operatorname{ker} \mathrm{L}=\left{y \in \mathrm{C}^2(I) \mid \mathrm{L} y=0\right} .$$

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

$$a_0(x) y^{\prime \prime}+a_1(x) y^{\prime}+a_2(x) y=0 .$$

$$y(x)=z(x) Y(x),$$
$z(x)$是新的未知函数。在(1.2.7)中替换它，我们得到

$$a_0(x) Y z^{\prime \prime}+\left[2 a_0(x) Y^{\prime}+a_1 Y\right] z^{\prime}+\left[a_0(x) Y^{\prime \prime}+a_1(x) Y^{\prime}+a_2(x) Y\right] z=0 .$$

$$a_0(x) Y u^{\prime}+\left[2 a_0(x) Y^{\prime}+a_1 Y\right] u=0 ;$$

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

$$u^{\prime}+\left(2 \frac{Y_1^{\prime}(x)}{Y_1(x)}+p(x)\right) u=0,$$

$$u(x)=C_1 \frac{e^{-\int p(x) \mathrm{d} x}}{Y_1^2(x)},$$

$$y(x)=C_1 Y_1(x) \int \frac{e^{-\int p(x) \mathrm{d} x}}{Y_1^2(x)} \mathrm{d} x .$$

## 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|M-544

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。

## 数学代写|常微分方程代写ordinary differential equation代考|THE METHOD OF VARIATION OF PARAMETERS

Except for $Y(x)$, formula (1.1.14) refers only to the coefficients of (1.1.1). Lagrange remarked that $Y(x)$ can be obtained in terms of these coefficients if we search it under the form
$$Y(x)=C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x},$$
that is, shaping it according to the general solution of the associated to (1.1.1) homogeneous equation. Introducing this in (1.1.1) yields
$$C^{\prime}(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}-p(x) C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}+p(x) C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}=f(x),$$
from which we deduce that $C(x)$ must satisfy
$$C^{\prime}(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}=f(x),$$
$$C^{\prime}(x)=f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} .$$
This is an equation considered at Sec.1.1. It follows that the general integral of (1.1.18) is written in the form
$$C(x)=K+\int f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} \mathrm{~d} x$$
In this expression, $\mathrm{K}$ is an arbitrary constant and the integral in the right member is a primitive of the function $f(x) \mathrm{e}^{\int p(x) \mathrm{d} x}$. Actually, we don’t need the general solution of (1.1.18) for our purpose; all we need is a particular solution, which can be found giving to $K$ an arbitrarily chosen value, e.g. $K=0$. With this, we get
$$Y(x)=\mathrm{e}^{-\int p(x) \mathrm{d} x} \int f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} \mathrm{~d} x .$$

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

Let us denote by $\mathrm{D}$ the operator indicating the derivative of first order of a function

$$\mathrm{D} \equiv \frac{\mathrm{d}}{\mathrm{d} x}$$
and by E the identity
$$\mathrm{E} y=y$$
Then L may be also expressed as
$$\mathrm{L} y=\mathrm{P}1(x, \mathrm{D}) y, \quad \mathrm{P}_1(x, \mathrm{D}) \equiv \mathrm{D}+p(x) \mathrm{E} .$$ The operator defined in (1.1.29) is a formal polynomial of first order in D and it is called a differential polynomial. Let now $\mathbf{y}=\left\lfloor y_j\right\rfloor{j=1, n}, \mathbf{f}=\left\lfloor f_j\right\rfloor_{j=1, n}$ be vector functions and assume that we must solve the vector equation
$$\mathrm{Ly} \equiv \dot{\mathbf{y}}+p(x) \mathbf{y}=\mathbf{f}, \quad p \in \mathrm{C}^0(I), \mathbf{f} \in\left(\mathrm{C}^0(I)\right)^n .$$
Writing (1.1.30) componentwisely, this means, in fact, that one has to solve $n$ uncoupled ODEs
$$\mathrm{L} y_j \equiv \dot{y}_j+p(x) y_j=f_j, \quad j=\overline{1, n} .$$

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|THE METHOD OF VARIATION OF PARAMETERS

$$Y(x)=C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x},$$

$$C^{\prime}(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}-p(x) C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}+p(x) C(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}=f(x),$$

$$C^{\prime}(x) \mathrm{e}^{-\int p(x) \mathrm{d} x}=f(x),$$

$$C^{\prime}(x)=f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} .$$

$$C(x)=K+\int f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} \mathrm{~d} x$$

$$Y(x)=\mathrm{e}^{-\int p(x) \mathrm{d} x} \int f(x) \mathrm{e}^{\int p(x) \mathrm{d} x} \mathrm{~d} x .$$

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

$$\mathrm{D} \equiv \frac{\mathrm{d}}{\mathrm{d} x}$$
E表示恒等式
$$\mathrm{E} y=y$$

$$\mathrm{L} y=\mathrm{P}1(x, \mathrm{D}) y, \quad \mathrm{P}1(x, \mathrm{D}) \equiv \mathrm{D}+p(x) \mathrm{E} .$$(1.1.29)中定义的算子是D中的一阶形式多项式，称为微分多项式。现在让$\mathbf{y}=\left\lfloor y_j\right\rfloor{j=1, n}, \mathbf{f}=\left\lfloor f_j\right\rfloor{j=1, n}$是向量函数假设我们必须解向量方程
$$\mathrm{Ly} \equiv \dot{\mathbf{y}}+p(x) \mathbf{y}=\mathbf{f}, \quad p \in \mathrm{C}^0(I), \mathbf{f} \in\left(\mathrm{C}^0(I)\right)^n .$$

$$\mathrm{L} y_j \equiv \dot{y}_j+p(x) y_j=f_j, \quad j=\overline{1, n} .$$

## 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|CRN18324

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。

## 数学代写|常微分方程代写ordinary differential equation代考|EQUATIONS OF THE FORM $y^{\prime}=f(x)$

This is the simplest form of (1.1.1). The solutions of this equation may be obviously regarded as primitives of $f$. Consequently, its general solution (integral) is
$$y(x)=\int f(x) d x+C,$$
where $\int f(x) d x$ is one of the primitives of $f$ and $C$ is an arbitrary constant. The representation (1.1.2) is obviously obtained by integrating both members of $y^{\prime}=f(x)$. If we wish to get the solution passing through the point $\left(x_0, y_0\right)$, where $x_0 \in I$, then it is convenient to choose $\int_{x_0}^x f(\xi) \mathrm{d} \xi$ among the primitives of $f$. Indeed, with this choice, the solution passes through $\left(x_0, y_0\right)$ if
$$C+\int_{x_0}^{x_0} f(\xi) \mathrm{d} \xi=y_0$$
therefore if $C=y_0$. This yields
$$y(x)=\int_{x_0}^x f(\xi) \mathrm{d} \xi+y_0 .$$

## 数学代写|常微分方程代写ordinary differential equation代考|THE LINEAR HOMOGENEOUS EQUATION

This equation is also a particular case of (1.1.1), where the free term is identically null, that is
$$y^{\prime}+p(x) y=0$$
Dividing by $y$ both terms of this equation, we immediately get
$$\frac{\mathrm{d}}{\mathrm{d} x}(\ln |y|)=-p(x) .$$
This means that $\ln |y|$ satisfies an equation of the previously considered type. Thus, the general solution of (1.1.6) is, by using directly (1.1.2),
$$\ln |y|=\widetilde{C}-\int p(x) \mathrm{d} x,$$
where $\widetilde{C}$ is an arbitrary constant and $\int p(x) \mathrm{d} x$ – one of the primitives of $p$. From (1.1.7) we see that $y$ is the general solution of (1.1.5) and is expressed by
$$y(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x},$$
with $C$ arbitrary constant.

Let us get back to the equation (1.1.1), in which the functions $f$ and $p$, defined on $I \subseteq \Re$, are not identically null. Suppose that we know a particular solution of (1.1.1), $Y(x)$ say, and let us perform the change of function
$$y(x)=z(x)+Y(x) .$$
Introducing this in (1.1.1) immediately involves
$$z^{\prime}+p(x) z+Y^{\prime}+p(x) Y=f(x) ;$$
thus, $z$ satisfies the homogeneous equation
$$z^{\prime}+p(x) z=0$$
which was studied at Sec.1.2 and whose general solution is
$$z(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x} .$$
Getting back to (1.1.10), we see that the general solution of (1.1.1) may be expressed in the form
$$y(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x}+Y(x),$$
where $Y(x)$ is a particular solution of the non-homogeneous equation (1.1.1). This form is very important, as it is characteristic for linear ODEs in general; we shall discuss it further.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|EQUATIONS OF THE FORM $y^{\prime}=f(x)$

$$y(x)=\int f(x) d x+C,$$

$$C+\int_{x_0}^{x_0} f(\xi) \mathrm{d} \xi=y_0$$

$$y(x)=\int_{x_0}^x f(\xi) \mathrm{d} \xi+y_0 .$$

## 数学代写|常微分方程代写ordinary differential equation代考|THE LINEAR HOMOGENEOUS EQUATION

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

$$\frac{\mathrm{d}}{\mathrm{d} x}(\ln |y|)=-p(x) .$$

$$\ln |y|=\widetilde{C}-\int p(x) \mathrm{d} x,$$

$$y(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x},$$

$$y(x)=z(x)+Y(x) .$$

$$z^{\prime}+p(x) z+Y^{\prime}+p(x) Y=f(x) ;$$

$$z^{\prime}+p(x) z=0$$

$$z(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x} .$$

$$y(x)=C \mathrm{e}^{-\int p(x) \mathrm{d} x}+Y(x),$$

## 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|MATH-UA262

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。

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

So far, mostly we have engaged ourselves in solving DEs, tacitly assuming that there always exists a solution. However, the theory of existence and uniqueness of solutions of the initial value problems is quite complex. We begin to develop this theory for the initial value problem
$$y^{\prime}=f(x, y), \quad y\left(x_0\right)=y_0,$$
where $f(x, y)$ will be assumed to be continuous in a domain $D$ containing the point $\left(x_0, y_0\right)$. By a solution of (7.1) in an interval $J$ containing $x_0$, we mean a function $y(x)$ satisfying (i) $y\left(x_0\right)=y_0$, (ii) $y^{\prime}(x)$ exists for all $x \in J$, (iii) for all $x \in J$ the points $(x, y(x)) \in D$, and (iv) $y^{\prime}(x)=f(x, y(x))$ for all $x \in J$.

For the initial value problem (7.1) later we shall prove that the continuity of the function $f(x, y)$ alone is sufficient for the existence of at least one solution in a sufficiently small neighborhood of the point $\left(x_0, y_0\right)$. However, if $f(x, y)$ is not continuous, then the nature of the solutions of $(7.1)$ is quite arbitrary. For example, the initial value problem
$$y^{\prime}=\frac{2}{x}(y-1), \quad y(0)=0$$
has no solution, while the problem
$$y^{\prime}=\frac{2}{x}(y-1), \quad y(0)=1$$
has an infinite number of solutions $y(x)=1+c x^2$, where $c$ is an arbitrary constant.

The use of integral equations to establish existence theorems is a standard device in the theory of DEs. It owes its efficiency to the smoothening properties of integration as contrasted with coarsening properties of differentiation. If two functions are close enough, their integrals must be close enough, whereas their derivatives may be far apart and may not even exist. We shall need the following result to prove the existence, uniqueness, and several other properties of the solutions of the initial value problem (7.1).

Theorem 7.1. Let $f(x, y)$ be continuous in the domain $D$, then any solution of (7.1) is also a solution of the integral equation
$$y(x)=y_0+\int_{x_0}^x f(t, y(t)) d t$$
and conversely.
Proof. Any solution $y(x)$ of the $\mathrm{DE} y^{\prime}=f(x, y)$ converts it into an identity in $x$, i.e., $y^{\prime}(x)=f(x, y(x))$. An integration of this equality yields
$$y(x)-y\left(x_0\right)=\int_{x_0}^x f(t, y(t)) d t .$$
Conversely, if $y(x)$ is any solution of $(7.2)$ then $y\left(x_0\right)=y_0$ and since $f(x, y)$ is continuous, differentiating (7.2) we find $y^{\prime}(x)=f(x, y(x))$.

## 数学代写|常微分方程代写ordinary differential equation代考|Picard’s Method of Successive Approximations

We shall solve the integral equation (7.2) by using the method of successive approximations due to Picard. For this, let $y_0(x)$ be any continuous function (we often pick $y_0(x) \equiv y_0$ ) which we assume to be the initial approximation of the unknown solution of (7.2), then we define $y_1(x)$ as
$$y_1(x)=y_0+\int_{x_0}^x f\left(t, y_0(t)\right) d t$$
We take this $y_1(x)$ as our next approximation and substitute this for $y(x)$ on the right side of (7.2) and call it $y_2(x)$. Continuing in this way, the $(m+1)$ st approximation $y_{m+1}(x)$ is obtained from $y_m(x)$ by means of the relation
$$y_{m+1}(x)=y_0+\int_{x_0}^x f\left(t, y_m(t)\right) d t, \quad m=0,1,2, \ldots$$
If the sequence $\left{y_m(x)\right}$ converges uniformly to a continuous function $y(x)$ in some interval $J$ containing $x_0$ and for all $x \in J$ the points $\left(x, y_m(x)\right) \in D$, then using Theorem 7.8 we may pass to the limit in both sides of (8.1), to obtain
$$y(x)=\lim {m \rightarrow \infty} y{m+1}(x)=y_0+\lim {m \rightarrow \infty} \int{x_0}^x f\left(t, y_m(t)\right) d t=y_0+\int_{x_0}^x f(t, y(t)) d t,$$
so that $y(x)$ is the desired solution.

# 常微分方程代写

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

$$y^{\prime}=f(x, y), \quad y\left(x_0\right)=y_0,$$

$$y^{\prime}=\frac{2}{x}(y-1), \quad y(0)=0$$

$$y^{\prime}=\frac{2}{x}(y-1), \quad y(0)=1$$

$$y(x)=y_0+\int_{x_0}^x f(t, y(t)) d t$$

$$y(x)-y\left(x_0\right)=\int_{x_0}^x f(t, y(t)) d t .$$

## 数学代写|常微分方程代写ordinary differential equation代考|Picard’s Method of Successive Approximations

$$y_1(x)=y_0+\int_{x_0}^x f\left(t, y_0(t)\right) d t$$

$$y_{m+1}(x)=y_0+\int_{x_0}^x f\left(t, y_m(t)\right) d t, \quad m=0,1,2, \ldots$$

$$y(x)=\lim {m \rightarrow \infty} y{m+1}(x)=y_0+\lim {m \rightarrow \infty} \int{x_0}^x f\left(t, y_m(t)\right) d t=y_0+\int_{x_0}^x f(t, y(t)) d t,$$

## 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|MATH376

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。

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

Suppose in the DE of first order (3.1), $M(x, y)=X_1(x) Y_1(y)$ and $N(x, y)=X_2(x) Y_2(y)$, so that it takes the form
$$X_1(x) Y_1(y)+X_2(x) Y_2(y) y^{\prime}=0 .$$
If $Y_1(y) X_2(x) \neq 0$ for all $(x, y) \in S$, then (4.1) can be written as an exact DE
$$\frac{X_1(x)}{X_2(x)}+\frac{Y_2(y)}{Y_1(y)} y^{\prime}=0$$
in which the variables are separated. Such a DE (4.2) is said to be separable. The solution of this exact equation is given by
$$\int \frac{X_1(x)}{X_2(x)} d x+\int \frac{Y_2(y)}{Y_1(y)} d y=c .$$
Here both the integrals are indefinite and constants of integration have been absorbed in $c$.

Example 4.1. The DE in Example 3.2 may be written as
$$\frac{1}{x}+\frac{1}{y(1-y)} y^{\prime}=0, \quad x y(1-y) \neq 0$$
for which (4.3) gives the solution $y=(1-c x)^{-1}$. Other possible solutions for which $x\left(y-y^2\right)=0$ are $x=0, y=0$, and $y=1$. However, the solution $y=1$ is already included in $y=(1-c x)^{-1}$ for $c=0$, and $x=0$ is not a solution, and hence all solutions of this DE are given by $y=0, y=(1-c x)^{-1}$.
A function $f(x, y)$ defined in a domain $D$ (an open connected set in $\mathbb{R}^2$ ) is said to be homogeneous of degree $k$ if for all real $\lambda$ and $(x, y) \in D$
$$f(\lambda x, \lambda y)=\lambda^k f(x, y)$$

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

Let in the DE (3.1) the functions $M$ and $N$ be $p_1(x) y-r(x)$ and $p_0(x)$, respectively, then it becomes
$$p_0(x) y^{\prime}+p_1(x) y=r(x),$$
which is a first-order linear DE. In (5.1) we shall assume that the functions $p_0(x), p_1(x), r(x)$ are continuous and $p_0(x) \neq 0$ in $J$. With these assumptions the DE (5.1) can be written as
$$y^{\prime}+p(x) y=q(x),$$
where $p(x)=p_1(x) / p_0(x)$ and $q(x)=r(x) / p_0(x)$ are continuous functions in $J$.
The corresponding homogeneous equation
$$y^{\prime}+p(x) y=0$$
obtained by taking $q(x) \equiv 0$ in (5.2) can be solved by separating the variables, i.e., $(1 / y) y^{\prime}+p(x)=0$, and now integrating it to obtain
$$y(x)=c \exp \left(-\int^x p(t) d t\right) .$$
In dividing (5.3) by $y$ we have lost the solution $y(x) \equiv 0$, which is called the trivial solution (for a linear homogeneous $\mathrm{DE} y(x) \equiv 0$ is always a solution). However, it is included in (5.4) with $c=0$.
If $x_0 \in J$, then the function
$$y(x)=y_0 \exp \left(-\int_{x_0}^x p(t) d t\right)$$
clearly satisfies the DE (5.3) in $J$ and passes through the point $\left(x_0, y_0\right)$. Thus, it is the solution of the initial value problem (5.3), (1.10).

To find the solution of the $\mathrm{DE}(5.2)$ we shall use the method of variation of parameters due to Lagrange. In (5.4) we assume that $c$ is a function of $x$, i.e.,
$$y(x)=c(x) \exp \left(-\int^x p(t) d t\right)$$

and search for $c(x)$ so that (5.6) becomes a solution of the DE (5.2). For this, substituting (5.6) into (5.2), we find
\begin{aligned} c^{\prime}(x) \exp \left(-\int^x p(t) d t\right) & -c(x) p(x) \exp \left(-\int^x p(t) d t\right) \ & +c(x) p(x) \exp \left(-\int^x p(t) d t\right)=q(x), \end{aligned}
which is the same as
$$c^{\prime}(x)=q(x) \exp \left(\int^x p(t) d t\right) .$$

# 常微分方程代写

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

$$X_1(x) Y_1(y)+X_2(x) Y_2(y) y^{\prime}=0 .$$

$$\frac{X_1(x)}{X_2(x)}+\frac{Y_2(y)}{Y_1(y)} y^{\prime}=0$$

$$\int \frac{X_1(x)}{X_2(x)} d x+\int \frac{Y_2(y)}{Y_1(y)} d y=c .$$

$$\frac{1}{x}+\frac{1}{y(1-y)} y^{\prime}=0, \quad x y(1-y) \neq 0$$
(4.3)给出了解决方案$y=(1-c x)^{-1}$。其他可能的解决方案$x\left(y-y^2\right)=0$是$x=0, y=0$和$y=1$。但是，$c=0$的解决方案$y=1$已经包含在$y=(1-c x)^{-1}$中，而$x=0$不是一个解决方案，因此该DE的所有解决方案都由$y=0, y=(1-c x)^{-1}$给出。

$$f(\lambda x, \lambda y)=\lambda^k f(x, y)$$

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

$$p_0(x) y^{\prime}+p_1(x) y=r(x),$$

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

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

$$y(x)=c \exp \left(-\int^x p(t) d t\right) .$$

$$y(x)=y_0 \exp \left(-\int_{x_0}^x p(t) d t\right)$$

$$y(x)=c(x) \exp \left(-\int^x p(t) d t\right)$$

\begin{aligned} c^{\prime}(x) \exp \left(-\int^x p(t) d t\right) & -c(x) p(x) \exp \left(-\int^x p(t) d t\right) \ & +c(x) p(x) \exp \left(-\int^x p(t) d t\right)=q(x), \end{aligned}

$$c^{\prime}(x)=q(x) \exp \left(\int^x p(t) d t\right) .$$

## 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|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。

## 数学代写|常微分方程代写ordinary differential equation代考|STABILITY OF SYSTEMS WITH LINEAR PART CRITICAL

The present section consists of two parts. In Section A, we consider time varying systems and in the second part we consider autonomous systems.
A. Time Varying Case
Let $g: R \times R^n \times\left[-\varepsilon_0, \varepsilon_0\right] \rightarrow R^n$ be $2 \pi$ periodic in $t$ and assume that $g$ is of class $C^2$ in $(x, \varepsilon)$. Suppose that $x^{\prime}=A x$ has a $2 \pi$-periodic solution $p(t)$ and suppose that
$$x^{\prime}=A x+\varepsilon g(t, x, \varepsilon)$$
has a continuous family of solutions $\psi(t, \varepsilon) \in \mathscr{P}_{2 \pi}$ with $\psi(t, 0)=p(t)$. To simplify matters, we specify the form of $A$ to be
$$A=\left[\begin{array}{ll} S & 0 \ 0 & C \end{array}\right], \quad S=\left[\begin{array}{cc} 0 & -N \ N & 0 \end{array}\right],$$
where $N$ is a positive integer and $C$ is an $(n-2) \times(n-2)$ constant matrix with no eigenvalues of the form $i M$ for any integer $M$.

The stability of the solution $\psi(t, \varepsilon)$ can be investigated using the linearization of $(6.1)$ about $\psi$, i.e.,
$$y^{\prime}=A y^{\prime}+\varepsilon g_x(t, \psi(t, \varepsilon), \varepsilon) y,$$
and Corollary 6.2.5. Let $Y(t, \varepsilon)$ be that fundamental matrix for this linear system which satisfies $Y(0, \varepsilon)=E$. Our problem is to determine whether or not all eigenvalues of $Y(2 \pi, \varepsilon)$ have magnitudes less than one for $\varepsilon$ near zero.
By the variation of constants formula, we can write
$$Y(t, \varepsilon)=e^{A t}+\varepsilon \int_0^t e^{A(t-s)} g_x(s, \psi(s, \varepsilon), \varepsilon) Y(s, \varepsilon) d s .$$
At $t=2 \pi$ we have $Y(2 \pi, \varepsilon)=e^{2 \pi R(\varepsilon)}$ for some $R(\varepsilon)$ so that
$$e^{2 \pi R(\varepsilon)}=e^{2 \pi A}\left{E+\varepsilon \int_0^{2 \pi} e^{-s A} g_x(s, \psi(s, r), r) Y\left(s, r_0\right) d s\right} .$$

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

We now study periodic systems of equations which can be decomposed into the form
\begin{aligned} x^{\prime} & =\varepsilon F(t, x, y, \varepsilon), \ y^{\prime} & =B y+\varepsilon G(t, x, y, \varepsilon), \end{aligned}
where $x \in R^n, y \in R^m, B$ is a constant $m \times m$ matrix and $F$ and $G$ are smooth functions defined on a neighborhood of $x=0, y=0, \varepsilon=0$ and are $2 \pi$ periodic in t. For $|y|$ and $|\varepsilon|$ small, we conjecture that $y$ has little effect on the first equation in (7.1). Indeed, it seems likely that the constant term in the Fourier series for $F$ provides a good approximation for $F(t, x, y, \varepsilon)$. Therefore, as an approximation we replace (7.1) by
\begin{aligned} x^{\prime} & =\varepsilon F_0(x), \ y^{\prime} & =B y \end{aligned}
where
$$F_0(x)=\frac{1}{2 \pi} \int_0^{2 \pi} F(u, x, 0,0) d u .$$
If (7.2) has a critical point $\left(x_0, 0\right)$ whose stability can be de ermined by lintarization, then we expect (7.1) to have a $2 \pi$-periodic solution which is near $\left(x_0, 0\right)$ and which has the same stability properties as $\left(x_0, 0\right)$. The following result shows that this approximate analysis is indeed valid.

Theorem 7.1. Let $F$ and $G$ be continuous in $(t, x, y, \varepsilon) \in R \times$ $B\left(x_0, h\right) \times B(h) \times\left[-\varepsilon_0, \varepsilon_0\right], 2 \pi$ periodic in $t$, and of class $C^2$ in $(x, y)$. Suppose that $F_y\left(l, x_0, 0,0\right)=0$. Let $\left(x_0, 0\right)$ be a critical point of $(7.2)$ such that all eigenvalues of the linearized system
$$x^{\prime}=\varepsilon \frac{\partial F_0}{\partial x}\left(x_0\right) x, \quad y^{\prime}=B y$$
have nonzero real parts for $\varepsilon \neq 0$. Then for $\varepsilon$ positive and sufficiently small, system (7.1) has a unique $2 \pi$-periodic solution $z(t, \varepsilon)=(x(t, \varepsilon), y(t, \varepsilon))$ in a neighborhood of $\left(x_0, 0\right)$ which is continuous in $(t, \varepsilon)$ and which satisfies $z(t, \varepsilon) \rightarrow\left(x_0, 0\right)$ as $\varepsilon \rightarrow 0^{+}$. Moreover, the stability properties of $z(t, \varepsilon)$ are the same as those of $\left(x_0, 0\right)$.

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|STABILITY OF SYSTEMS WITH LINEAR PART CRITICAL

A.时变情况

$$x^{\prime}=A x+\varepsilon g(t, x, \varepsilon)$$

$$A=\left[\begin{array}{ll} S & 0 \ 0 & C \end{array}\right], \quad S=\left[\begin{array}{cc} 0 & -N \ N & 0 \end{array}\right],$$

$$y^{\prime}=A y^{\prime}+\varepsilon g_x(t, \psi(t, \varepsilon), \varepsilon) y,$$

$$Y(t, \varepsilon)=e^{A t}+\varepsilon \int_0^t e^{A(t-s)} g_x(s, \psi(s, \varepsilon), \varepsilon) Y(s, \varepsilon) d s .$$

$$e^{2 \pi R(\varepsilon)}=e^{2 \pi A}\left{E+\varepsilon \int_0^{2 \pi} e^{-s A} g_x(s, \psi(s, r), r) Y\left(s, r_0\right) d s\right} .$$

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

\begin{aligned} x^{\prime} & =\varepsilon F(t, x, y, \varepsilon), \ y^{\prime} & =B y+\varepsilon G(t, x, y, \varepsilon), \end{aligned}

\begin{aligned} x^{\prime} & =\varepsilon F_0(x), \ y^{\prime} & =B y \end{aligned}

$$F_0(x)=\frac{1}{2 \pi} \int_0^{2 \pi} F(u, x, 0,0) d u .$$

$$x^{\prime}=\varepsilon \frac{\partial F_0}{\partial x}\left(x_0\right) x, \quad y^{\prime}=B y$$

## 有限元方法代写

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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|MATH-UA 262

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## 数学代写|常微分方程代写ordinary differential equation代考|POINCARÉ-BENDIXSON THEORY

We shall construct Jordan curves with the aid of transversals. A transversal with respect to the continuous function $f: R^2 \rightarrow R^2$ is a closed line segment $L$ in $R^2$ such that every point of $L$ is a regular point and for each point $\xi \in L$, the vector $f(\xi)$ is not parallel to the direction of the line segment $L$. We note that since $f$ is continuous, given any regular point $\xi \in R^2$ and any direction $\eta \in R^2$ which is not parallel to $f(\xi)$ [i.e., $\eta \neq \alpha f(\xi)$ for any nonzero constant $\alpha \in R$, there is a transversal through $\xi$ in the direction of $\eta$. Note also that if an orbit of (A) meets a transversal $L$, it must cross $L$. Moreover, all such crossings of $L$ are in the same direction. A deeper property of transversals is summarized in the following result.

Lemma 2.1. If $\xi_0$ is an interior point of a transversal $L$, then for any $\varepsilon>0$ there is a $\delta>0$ such that any orbit passing through the ball $B\left(\xi_0, \delta\right)$ at $t=0$ must cross $L$ at some time $t \in(-\varepsilon, \varepsilon)$.

Proof. Suppose the transversal $L$ has direction $\eta=\left(\eta_1, \eta_2\right)^{\top}$. Then points $x=\left(x_1, x_2\right)^{\mathrm{T}}$ of $L$ will satisfy an equation of the form
$$g(x) \triangleq a_1 x_1+a_2 x_2-c=0$$
where $c$ is a constant and $a=\left(a_1, a_2\right)^{\top}$ is a vector such that $a^{\top} \eta=0$ and $|a| \neq 0$. Let $\phi(t, \xi)$ be the solution of $(A)$ such that $\phi(0, \xi)=\xi$ and define $G$ by
$$G(t, \xi)=g(\phi(t, \xi))$$

Then $G\left(0, \xi_0\right)=0$ since $\xi_0 \in L$ and
$$\frac{\partial G}{\partial t}\left(0, \xi_0\right)=a^{\top} f\left(\xi_0\right) \neq 0$$
since $L$ is a transversal. By the implicit function theorem, there is a $C^1$ function $t: B\left(\xi_0, \delta\right) \rightarrow R$ for some $\delta>0$ such that $t\left(\xi_0\right)=0$ and $G(t(\xi), \xi) \equiv 0$. By possibly reducing the size of $\delta$, it can be assumed that $|t(\xi)|<\varepsilon$ when $\xi \in B\left(\xi_0, \delta\right)$. Hence $\phi(t, \xi)$ will cross $L$ at $t(\xi)$ and $-\varepsilon<t(\xi)<\varepsilon$.

## 数学代写|常微分方程代写ordinary differential equation代考|THE LEVINSON-SMITH THEOREM

The purpose of this section is to prove a result of Levinson and Smith concerning limit cycles of Lienard equations of the form
$$x^{\prime \prime}+f(x) x^{\prime}+g(x)=0$$
when $f$ and $y$ satisfy the following assumptions:
$f: R \rightarrow R \quad$ is even and continuous, and
$g: R \rightarrow R \quad$ is odd, is in $C^1(R)$, and $x g(x)>0$ for all $x \neq 0$;
there is a constant $a>0$ such that $F(x) \triangleq \int_0^x f(s) d s<0$ $$\begin{gathered} \text { on } 00 \text { on } x>a \text {, and } f(x)>0 \text { on } x>a ; \ G(x) \triangleq \int_0^x g(s) d s \rightarrow \infty \text { as }|x| \rightarrow \infty \text { and } F(x) \rightarrow \infty \text { as } x \rightarrow \infty . \end{gathered}$$
We now prove the following result.

Theorem 3.1. If Eq. (3.1) satisfies hypotheses (3.2)-(3.4), then there is a nonconstant, orbitally stable periodic solution $p(t)$ of Eq. (3.1). This periodic solution is unique up to translations $p(t+\tau), \tau \in R$.

Proof. Under the change of variables $y=x^{\prime}+F(x)$, Eq. (3.1) is equivalent to
$$x^{\prime}=y-F(x), \quad y^{\prime}=-g(x) .$$
The coefficients of (3.5) are smooth enough to ensure local existence and uniqueness of the initial value problem determined by (3.5). Hence, existence and uniqueness conditions are also satisfied by a corresponding initial value problem determined by (3.1).
Now define a Lyapunov function for (3.5) by
$$v(x, y)=y^2 / 2+G(x) \text {. }$$
The derivative of $v$ with respect to $t$ along solutions of Eq. (3.5) is given by
$$d v / d t=v_{(3.5)}^{\prime}(x, y)=-g(x) F(x)$$

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|POINCARÉ-BENDIXSON THEORY

$$g(x) \triangleq a_1 x_1+a_2 x_2-c=0$$

$$G(t, \xi)=g(\phi(t, \xi))$$

$$\frac{\partial G}{\partial t}\left(0, \xi_0\right)=a^{\top} f\left(\xi_0\right) \neq 0$$

## 数学代写|常微分方程代写ordinary differential equation代考|THE LEVINSON-SMITH THEOREM

$$x^{\prime \prime}+f(x) x^{\prime}+g(x)=0$$

$f: R \rightarrow R \quad$是偶连续的，且
$g: R \rightarrow R \quad$是奇数，是在$C^1(R)$, $x g(x)>0$是所有的$x \neq 0$;

$$S_N^{\prime}(t)=A S_{N-1}(t)=S_{N-1}(t) A$$

## 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。