如果你也在 怎样代写微分方程differential equation这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。微分方程differential equation在数学中，是将一个或多个未知函数及其导数联系起来的方程。在应用中，函数通常代表物理量，导数代表其变化率，而微分方程则定义了两者之间的关系。这种关系很常见；因此，微分方程在许多学科，包括工程、物理学、经济学和生物学中发挥着突出作用。
微分方程differential equation研究主要包括研究其解（满足每个方程的函数集合），以及研究其解的性质。只有最简单的微分方程可以用明确的公式求解；然而，一个给定的微分方程的解的许多属性可以在不精确计算的情况下确定。
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数学代写|微分方程代写differential equation代考|Analytical solutions of linear state space equations
Linear state space models are the most widely used linear differential equations in many fields. They provide an alternative description format for linear differential equations with constant coefficients. For instance, in control science, linear state space models are used to describe linear dynamical systems. In this section, the mathematical form of a state space model is presented, and analytical solution methods are illustrated.
Definition 2.18. A linear state space model can be expressed as
$$
\left{\begin{array}{l}
\boldsymbol{x}^{\prime}(t)=\boldsymbol{A} \boldsymbol{x}(t)+\boldsymbol{B} \boldsymbol{u}(t) \
\boldsymbol{y}(t)=\boldsymbol{C x}(t)+\boldsymbol{D} \boldsymbol{u}(t)
\end{array}\right.
$$
where $\boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C}, \boldsymbol{D}$ are constant matrices. Vector $\boldsymbol{x}(t)$ is referred to as a state variable vector. The initial state vector is known $\boldsymbol{x}\left(t_{0}\right)=\boldsymbol{x}_{0}$. Signals $\boldsymbol{u}(t)$ and $\boldsymbol{y}(t)$ are regarded respectively as the input and output of the system.
The first equation in (2.5.1) can be rewritten as
$$
\boldsymbol{x}^{\prime}(t)-\boldsymbol{A} \boldsymbol{x}(t)=\boldsymbol{B} \boldsymbol{u}(t)
$$
Multiplying both sides by $\mathrm{e}^{-A t}$, it is found that
$$
\mathrm{e}^{-\boldsymbol{A} t} \boldsymbol{\chi}^{\prime}(t)-\mathrm{e}^{-\boldsymbol{A} t} \boldsymbol{A} \boldsymbol{x}(t)=\mathrm{e}^{-\boldsymbol{A} t} \boldsymbol{B} \boldsymbol{u}(t)
$$
Since $\boldsymbol{A} \mathrm{e}^{-\boldsymbol{A t}}=\mathrm{e}^{-\boldsymbol{A t} \boldsymbol{A}}$, the left side happens to be the derivative formula
$$
\frac{\mathrm{d}}{\mathrm{d} t}\left[\mathrm{e}^{-\boldsymbol{A} t} \boldsymbol{x}(t)\right]=\mathrm{e}^{-\boldsymbol{A} t} \boldsymbol{B} \boldsymbol{u}(t)
$$
数学代写|微分方程代写differential equation代考|Direct solutions of state space models
Function dsolve() can be used to solve the state space equations directly. The difficulties of using such a function appear in constructing a symbolic function matrix $\boldsymbol{X}(t)$. A MATLAB any_matrix () function is written to define immediately any symbolic function matrix. The listing of the function is as follows. The syntaxes of the function will be demonstrated later through examples.
function A=any_matrix (nn,sA, varargin) \% generate arbitrary matrix function $\mathrm{v}=$ varargin; $\mathrm{n}=\mathrm{nn}(1)$;
if length $(\mathrm{nn})==1, \mathrm{~m}=\mathrm{n}$; else, $\mathrm{m}=\mathrm{nn}(2)$; end
$\mathrm{s}={ }^{\prime \prime} ; \mathrm{k}=$ length (v); $\mathrm{K}=0$; if $\mathrm{n}==1 \mathrm{|} \mathrm{m}==1, \mathrm{~K}=1$; end
if $\mathrm{k}>0, \mathrm{~s}=$ ‘ (‘; for $i=1: \mathrm{k}, \mathrm{s}=[\mathrm{s}$, , $\operatorname{char}(\mathrm{v}{i})]$; end
$$
\left.s(2)=[] ; s=[s,)^{\prime}\right] \text {; }
$$
end
for $i=1: \mathrm{n}$, for $j=1: \mathrm{m} \%$ loop structure to handle each element individually
if $K==0, s t r=[s A$ int2str(i), int2str $(j) s]$;
else, str=[sA int2str(i*j) s]; end
eval ([‘syms’, str]); $\operatorname{eval}([‘ A(i, j)=$, str ‘, ” $])$;
end, end
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|Solution of Sylvester differential equation
Sylvester differential equation is a special form of matrix differential equations. The definition of the equation is presented first in the section, and then three solution methods are explored.
Definition 2.20. Matrix Sylvester differential equation is expressed as ${ }^{[43]}$
$$
\boldsymbol{X}^{\prime}(t)=\boldsymbol{A} \boldsymbol{X}(t)+\boldsymbol{X}(t) \boldsymbol{B}, \quad \boldsymbol{X}(0)=\boldsymbol{X}{0} $$ where $\boldsymbol{A} \in \mathscr{R}^{n \times n}, \boldsymbol{B} \in \mathscr{R}^{m \times m}$, and $\boldsymbol{X}, \boldsymbol{X}{0} \in \mathscr{R}^{n \times m}$.
Theorem 2.13. The analytical solution of the matrix Sylvester differential equation ${ }^{[43]}$ is $\boldsymbol{X}(t)=\mathrm{e}^{\boldsymbol{A} t} \boldsymbol{X}_{0} \mathrm{e}^{\boldsymbol{B} t}$.
微分方程代写
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|ANALYTICAL SOLUTIONS OF LINEAR STATE SPACE EQUATIONS
线性状态空间模型是许多领域中应用最广泛的线性微分方程。它们为具有常数系数的线性微分方程提供了另一种描述格式。例如，在控制科学中，线性状态空间模型用于描述线性动力系统。在本节中，介绍了状态空间模型的数学形式，并说明了解析求解方法。
定义 2.18。线性状态空间模型可以表示为
$$
\left{\begin{array}{l}
\boldsymbol{x}^{\prime}(t)=\boldsymbol{A} \boldsymbol{x}(t)+\boldsymbol{B} \boldsymbol{u}(t) \
\boldsymbol{y}(t)=\boldsymbol{C x}(t)+\boldsymbol{D} \boldsymbol{u}(t)
\end{array}\right.
$$
where $\boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C}, \boldsymbol{D}$ are constant matrices. Vector $\boldsymbol{x}(t)$ is referred to as a state variable vector. The initial state vector is known $\boldsymbol{x}\left(t_{0}\right)=\boldsymbol{x}_{0}$. Signals $\boldsymbol{u}(t)$ and $\boldsymbol{y}(t)$ are regarded respectively as the input and output of the system.
The first equation in (2.5.1) can be rewritten as
$$
\boldsymbol{x}^{\prime}(t)-\boldsymbol{A} \boldsymbol{x}(t)=\boldsymbol{B} \boldsymbol{u}(t)
$$
Multiplying both sides by $\mathrm{e}^{-A t}$, it is found that
$$
\mathrm{e}^{-\boldsymbol{A} t} \boldsymbol{\chi}^{\prime}(t)-\mathrm{e}^{-\boldsymbol{A} t} \boldsymbol{A} \boldsymbol{x}(t)=\mathrm{e}^{-\boldsymbol{A} t} \boldsymbol{B} \boldsymbol{u}(t)
$$
Since $\boldsymbol{A} \mathrm{e}^{-\boldsymbol{A t}}=\mathrm{e}^{-\boldsymbol{A t} \boldsymbol{A}}$, the left side happens to be the derivative formula
$$
\frac{\mathrm{d}}{\mathrm{d} t}\left[\mathrm{e}^{-\boldsymbol{A} t} \boldsymbol{x}(t)\right]=\mathrm{e}^{-\boldsymbol{A} t} \boldsymbol{B} \boldsymbol{u}(t)
$$
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|DIRECT SOLUTIONS OF STATE SPACE MODELS
函数 dsolve可用于直接求解状态空间方程。使用这种函数的困难出现在构造符号函数矩阵时X(吨). 一个 MATLAB any_matrix编写函数以立即定义任何符号函数矩阵。函数列表如下。稍后将通过示例演示该函数的语法。
函数 A=any_matrixnn,s一种,v一种r一种rG一世n\% 生成任意矩阵函数v=可变精；n=nn(1);
如果长度=any_matrix (nn,sA, varargin) \% generate arbitrary matrix function $\mathrm{v}=$ varargin; $\mathrm{n}=\mathrm{nn}(1)$;
if length $(\mathrm{nn})==1, \mathrm{~m}=\mathrm{n}$; else, $\mathrm{m}=\mathrm{nn}(2)$; end
$\mathrm{s}={ }^{\prime \prime} ; \mathrm{k}=$ length (v); $\mathrm{K}=0$; if $\mathrm{n}==1 \mathrm{|} \mathrm{m}==1, \mathrm{~K}=1$; end
if $\mathrm{k}>0, \mathrm{~s}=$ ‘ (‘; for $i=1: \mathrm{k}, \mathrm{s}=[\mathrm{s}$, , $\operatorname{char}(\mathrm{v}{i})]$; end
$$
\left.s(2)=[] ; s=[s,)^{\prime}\right] \text {; }
$$
end
for $i=1: \mathrm{n}$, for $j=1: \mathrm{m} \%$ loop structure to handle each element individually
if $K==0, s t r=[s A$ int2str(i), int2str $(j) s]$;
else, str=[sA int2str(i*j) s]; end
eval ([‘syms’, str]); $\operatorname{eval}([‘ A(i, j)=$, str ‘, ” $])$;
结束，结束
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|SOLUTION OF SYLVESTER DIFFERENTIAL EQUATION
西尔维斯特微分方程是矩阵微分方程的一种特殊形式。本节首先介绍方程的定义，然后探讨三种求解方法。
定义 2.20。矩阵西尔维斯特微分方程表示为[43]
$$
\boldsymbol{X}^{\prime}(t)=\boldsymbol{A} \boldsymbol{X}(t)+\boldsymbol{X}(t) \boldsymbol{B}, \quad \boldsymbol{X}(0)=\boldsymbol{X}{0} $$ where $\boldsymbol{A} \in \mathscr{R}^{n \times n}, \boldsymbol{B} \in \mathscr{R}^{m \times m}$, and $\boldsymbol{X}, \boldsymbol{X}{0} \in \mathscr{R}^{n \times m}$.
Theorem 2.13. The analytical solution of the matrix Sylvester differential equation ${ }^{[43]}$ is $\boldsymbol{X}(t)=\mathrm{e}^{\boldsymbol{A} t} \boldsymbol{X}_{0} \mathrm{e}^{\boldsymbol{B} t}$.
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