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# 数学代写|偏微分方程代考Partial Differential Equations代写|LINEAR FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS

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## 数学代写|偏微分方程代考Partial Differential Equations代写|LINEAR FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS

Partial differential equations of the form
$$P p+Q q=R$$
where $P, Q$, and $R$ are functions of $x, y, z$ or constants, $p$ denotes $\frac{\partial z}{\partial x}$, and $q$ denotes $\frac{\partial z}{\partial y}$ is called the Lagrange linear equation of the first order.

To solve the Lagrange linear equation $P p+Q q=R$, we will follow the following algorithm.
Step 1: Form the auxiliary equation $\frac{d x}{P}=\frac{d y}{Q}=\frac{d z}{R}$.
Step 2: The auxiliary equations can be solved using the grouping method or the multiplier method (described below) or both to get two independent solutions of auxiliary equations, denoted by
$u(x, y, z)=c_1, v(x, y, z)=c_2$; where $c_1$ and $c_2$ are arbitrary constants.

Step 3: Then $F(u, v)=0$ or $u=f(v)$ is the general solution of the given equation
$$P p+Q q=R$$

• Grouping Method
In this method, we compare any two functions which make integration possible. In other words, to complete the first two fractions, the remaining third variable must be absent from them or it is possible to eliminate it by appropriate operations.
• Multipliers Method
In this method, we find two sets of multiplier $l, m, n$ and $l^{\prime}, m^{\prime}$, and $n^{\prime}$ either constant or functions of $x, y, z$ such that
$$l P+m Q+n R=0 \text { and } l^{\prime} P+m^{\prime} Q+n^{\prime} R=0$$
Or the selection makes the integration possible.

## 数学代写|偏微分方程代考Partial Differential Equations代写|CHARPIT MFTHOD

To find the complete solution of the first-order non-linear partial differential equation of the form
$$f(x, y, z, p, q)=0$$
The main concept of Charpit method is the introduction of another first-order partial differential equation of the form
$$F(x, y, z, p, q)=0$$
Solve the above two equations for $p$ and $q$, and substitute in
$$d z=p(x, y, z, a) d x+q(x, y, z, a) d y$$

The solution of the above equation, if it exists, is the complete solution of the equation $f(x, y, z, p, q)=0$.

Now, the main thing is to determine the $F(x, y, z, p, q)=0$ which is compatible with the equation $d z=p(x, y, z, a) d x+q(x, y, z, a) d y$.
The necessary and sufficient condition is
$$\begin{gathered} \frac{\partial(f, F)}{\partial(x, p)}+p \frac{\partial(f, F)}{\partial(z, p)}+\frac{\partial(f, F)}{\partial(y, q)}+q \frac{\partial(f, F)}{\partial(z, q)}=0 \ \therefore\left(\frac{\partial f}{\partial x} \frac{\partial F}{\partial p}-\frac{\partial f}{\partial p} \frac{\partial F}{\partial x}\right)+p\left(\frac{\partial f}{\partial z} \frac{\partial F}{\partial p}-\frac{\partial f}{\partial p} \frac{\partial F}{\partial z}\right)+\left(\frac{\partial f}{\partial y} \frac{\partial F}{\partial q}-\frac{\partial f}{\partial q} \frac{\partial F}{\partial y}\right) \ +q\left(\frac{\partial f}{\partial z} \frac{\partial F}{\partial q}-\frac{\partial f}{\partial q} \frac{\partial F}{\partial z}\right)=0 \ \therefore f_p \frac{\partial F}{\partial x}+f_q \frac{\partial F}{\partial y}+\left(p f_p+q f_q\right) \frac{\partial F}{\partial z}-\left(f_x+p f_z\right) \frac{\partial F}{\partial p}-\left(f_y+q f_z\right) \frac{\partial F}{\partial q}=0 \end{gathered}$$
The above equation is a linear partial differential equation.
So, the auxiliary equation is
$$\frac{d x}{f_p}=\frac{d y}{f_q}=\frac{d z}{p f_p+q f_q}=\frac{d p}{-\left(f_x+p f_z\right)}=\frac{d q}{-\left(f_y+q f_z\right)} .$$
The above equation is called the Charpit auxiliary equation.
The Charpit method is illustrated through the examples as follows.

# 偏微分方程代写

## 代写|偏微分方程代考Partial Differential Equations代写|LINEAR FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS

$$P p+Q q=R$$

$u(x, y, z)=c_1, v(x, y, z)=c_2$;其中$c_1$和$c_2$是任意常数。

$$P p+Q q=R$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|CHARPIT MFTHOD

$$l P+m Q+n R=0 \text { and } l^{\prime} P+m^{\prime} Q+n^{\prime} R=0$$

$$f(x, y, z, p, q)=0$$
Charpit方法的主要概念是引入另一种形式的一阶偏微分方程
$$F(x, y, z, p, q)=0$$

$$d z=p(x, y, z, a) d x+q(x, y, z, a) d y$$

$$\begin{gathered} \frac{\partial(f, F)}{\partial(x, p)}+p \frac{\partial(f, F)}{\partial(z, p)}+\frac{\partial(f, F)}{\partial(y, q)}+q \frac{\partial(f, F)}{\partial(z, q)}=0 \ \therefore\left(\frac{\partial f}{\partial x} \frac{\partial F}{\partial p}-\frac{\partial f}{\partial p} \frac{\partial F}{\partial x}\right)+p\left(\frac{\partial f}{\partial z} \frac{\partial F}{\partial p}-\frac{\partial f}{\partial p} \frac{\partial F}{\partial z}\right)+\left(\frac{\partial f}{\partial y} \frac{\partial F}{\partial q}-\frac{\partial f}{\partial q} \frac{\partial F}{\partial y}\right) \ +q\left(\frac{\partial f}{\partial z} \frac{\partial F}{\partial q}-\frac{\partial f}{\partial q} \frac{\partial F}{\partial z}\right)=0 \ \therefore f_p \frac{\partial F}{\partial x}+f_q \frac{\partial F}{\partial y}+\left(p f_p+q f_q\right) \frac{\partial F}{\partial z}-\left(f_x+p f_z\right) \frac{\partial F}{\partial p}-\left(f_y+q f_z\right) \frac{\partial F}{\partial q}=0 \end{gathered}$$

$$\frac{d x}{f_p}=\frac{d y}{f_q}=\frac{d z}{p f_p+q f_q}=\frac{d p}{-\left(f_x+p f_z\right)}=\frac{d q}{-\left(f_y+q f_z\right)} .$$

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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。