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|Simple explicit differential equation sets
In this section, two differential equations are demonstrated and we show how to convert them into first-order explicit differential equations. Suppose the two equations can be written as the following differential equations:
$$
\left{\begin{array}{l}
x^{(m)}(t)=f\left(t, x(t), x^{\prime}(t), \ldots, x^{(m-1)}(t), y(t), y^{\prime}(t), \ldots, y^{(n-1)}(t)\right) \
y^{(n)}(t)=g\left(t, x(t), x^{\prime}(t), \ldots, x^{(m-1)}(t), y(t), y^{\prime}(t), \ldots, y^{(n-1)}(t)\right)
\end{array}\right.
$$
where each equation may contain the explicit form of the highest-order derivative of one unknown function. The state variables can still be selected as $x_{1}(t)=x(t), x_{2}(t)=$ $x^{\prime}(t), \ldots, x_{m}(t)=\chi^{(m-1)}(t)$. Then one may continue selecting the state variables as $x_{m+1}=y(t), x_{m+2}=y^{\prime}(t), \ldots, x_{m+n}(t)=y^{(n-1)}(t)$. In this case, the original differential equations can be converted into
$$
\left{\begin{array}{l}
x_{1}^{\prime}(t)=x_{2}(t) \
\quad \vdots \
x_{m}^{\prime}(t)=f\left(t, x_{1}(t), x_{2}(t), \ldots, x_{m+n}(t)\right) \
x_{m+1}^{\prime}(t)=x_{m+2}(t) \
\quad \vdots \
x_{m+n}^{\prime}(t)=g\left(t, x_{1}(t), x_{2}(t), \ldots, x_{m+n}(t)\right)
\end{array}\right.
$$
The initial values of the state variables can also be set accordingly. The expected first-order explicit differential equations can then be established. An example will be given next to demonstrate the conversion and solution process.
数学代写|微分方程代写differential equation代考|Limitations with fixed-step methods
In conventional numerical analysis courses, various fixed-step algorithms are presented. The main reason is that fixed-step algorithms have neat mathematical formulas, and are easy to teach and numerically implement. While in real applications, fixed-step methods are not practical. The Apollo satellite example is used as an example to illustrate the limitations of the fixed-step algorithms.
Example 4.16. Solve the Apollo satellite equation again with the fixed-step fourthorder Runge-Kutta algorithm.
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|Simple implicit differential equations
In the previously studied examples, the explicit form of the highest-order derivative of the unknown function could be found. It was easy to build up the first-order explicit differential equation model. In this section, complicated differential equation sets are discussed, where in each equation the highest-order derivatives of two unknown functions appear simultaneously. Algebraic equation solution must be carried out in describing the first-order explicit differential equations. Examples are given to demonstrate the conversion and solving methods.
Example 4.17. Consider the following differential equations, if $x_{1}(0)=x_{2}(0)=0$,
$$
\left{\begin{array}{l}
x_{1}^{\prime}(t) \sin x_{1}(t)+x_{2}^{\prime}(t) \cos x_{2}(t)+x_{1}(t)=1, \
-x_{1}^{\prime}(t) \cos x_{2}(t)+x_{2}^{\prime}(t) \sin x_{1}(t)+x_{2}(t)=0 .
\end{array}\right.
$$
Find their numerical solution.
微分方程代写
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|SIMPLE EXPLICIT DIFFERENTIAL EQUATION SETS
在本节中，演示了两个微分方程，并展示了如何将它们转换为一阶显式微分方程。假设这两个方程可以写成以下微分方程：
$$
\left{\begin{array}{l}
x^{(m)}(t)=f\left(t, x(t), x^{\prime}(t), \ldots, x^{(m-1)}(t), y(t), y^{\prime}(t), \ldots, y^{(n-1)}(t)\right) \
y^{(n)}(t)=g\left(t, x(t), x^{\prime}(t), \ldots, x^{(m-1)}(t), y(t), y^{\prime}(t), \ldots, y^{(n-1)}(t)\right)
\end{array}\right.
$$
where each equation may contain the explicit form of the highest-order derivative of one unknown function. The state variables can still be selected as $x_{1}(t)=x(t), x_{2}(t)=$ $x^{\prime}(t), \ldots, x_{m}(t)=\chi^{(m-1)}(t)$. Then one may continue selecting the state variables as $x_{m+1}=y(t), x_{m+2}=y^{\prime}(t), \ldots, x_{m+n}(t)=y^{(n-1)}(t)$. In this case, the original differential equations can be converted into
$$
\left{\begin{array}{l}
x_{1}^{\prime}(t)=x_{2}(t) \
\quad \vdots \
x_{m}^{\prime}(t)=f\left(t, x_{1}(t), x_{2}(t), \ldots, x_{m+n}(t)\right) \
x_{m+1}^{\prime}(t)=x_{m+2}(t) \
\quad \vdots \
x_{m+n}^{\prime}(t)=g\left(t, x_{1}(t), x_{2}(t), \ldots, x_{m+n}(t)\right)
\end{array}\right.
$$
状态变量的初始值也可以相应设置。然后可以建立预期的一阶显式微分方程。下面将给出一个例子来演示转换和求解过程。
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|LIMITATIONS WITH FIXED-STEP METHODS
在传统的数值分析课程中，介绍了各种固定步长算法。主要原因是固定步长算法具有简洁的数学公式，易于教授和数值实现。而在实际应用中，固定步长方法并不实用。以阿波罗卫星为例说明固定步长算法的局限性。
例 4.16。使用固定步长四阶 Runge-Kutta 算法再次求解 Apollo 卫星方程。
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|SIMPLE IMPLICIT DIFFERENTIAL EQUATIONS
在前面研究的例子中，可以找到未知函数的最高阶导数的显式形式。建立一阶显式微分方程模型很容易。在本节中，将讨论复杂的微分方程组，其中在每个方程中两个未知函数的最高阶导数同时出现。描述一阶显式微分方程必须进行代数方程解。举例说明转换和求解方法。
例 4.17。考虑以下微分方程，如果X1(0)=X2(0)=0,
$$
\左{X1′(吨)没有X1(吨)+X2′(吨)某物X2(吨)+X1(吨)=1, −X1′(吨)某物X2(吨)+X2′(吨)没有X1(吨)+X2(吨)=0.\对。
$$
找到他们的数值解。
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