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|Equations containing the square of the highest-order derivative
Suppose the highest order term of the unknown function appears in the square form
$$
\left[y^{(n)}(t)\right]^{2}=f\left(t, y(t), y^{\prime}(t), \ldots, y^{(n-1)}(t)\right)
$$
and the initial values $y\left(t_{0}\right), y^{\prime}\left(t_{0}\right), \ldots, y^{(n-1)}\left(t_{0}\right)$ are given. As before, the state variables $x_{1}(t)=y(t), x_{2}(t)=y^{\prime}(t), \ldots, x_{n}(t)=y^{(n-1)}(t)$ can be selected first, and taking the square root of the last term, two different sets of first-order explicit differential equations can be created
$$
\left{\begin{aligned}
x_{1}^{\prime}(t) &=x_{2}(t) \
& \vdots \
x_{n-1}^{\prime}(t) &=x_{n}(t) \
x_{n}^{\prime}(t) &=\sqrt{f\left(t, y(t), y^{\prime}(t), \ldots, y^{(n-1)}(t)\right)}
\end{aligned}\right.
$$
and
$$
\left{\begin{aligned}
x_{1}^{\prime}(t) &=x_{2}(t) \
& \vdots \
x_{n-1}^{\prime}(t) &=x_{n}(t) \
x_{n}^{\prime}(t) &=-\sqrt{f\left(t, y(t), y^{\prime}(t), \ldots, y^{(n-1)}(t)\right)} .
\end{aligned}\right.
$$
The two state space models comprise the original differential equation. The initial values of the states are
$$
\boldsymbol{x}\left(t_{0}\right)=\left[y\left(t_{0}\right), y^{\prime}\left(t_{0}\right), \ldots, y^{(n-1)}\left(t_{0}\right)\right]^{\mathrm{T}}
$$
It can be seen that the two differential equation systems can both be solved directly with MATLAB solvers. Therefore, with anonymous or MATLAB functions, the two equation sets can be described so that they can be solved numerically. Both solutions satisfy the original differential equation.
数学代写|微分方程代写differential equation代考|Equations containing odd powers
In real applications there are even differential equations of complicated form. For instance, the odd power of the highest-order derivative term may exist:
$$
\left[y^{n}(t)\right]^{2 k+1}=f\left(t, x_{1}(t), x_{2}(t), \ldots, x_{n}(t)\right) .
$$
In the real domain, manipulation of such differential equations looks simpler, compared with the square counterpart, since the root may not have the phenomenon of multiple solutions. The state variables can be selected as usual, for instance, $x_{1}(t)=$ $y(t), x_{2}(t)=y^{\prime}(t), \ldots, x_{n}(t)=y^{(n-1)}(t)$. Finally, the first-order explicit differential equation can be written as
$$
\boldsymbol{x}^{\prime}(t)=\left[\begin{array}{c}
\chi_{2}(t) \
\vdots \
x_{n-1}(t) \
\sqrt[2 k+1]{f\left(t, x_{1}(t), \chi_{2}(t), \ldots, x_{n}(t)\right)}
\end{array}\right] .
$$
The next two examples are created by the author based on a given function $y(t)=$ $\mathrm{e}^{-t}$. If one wants to find the analytical solution in a usual way, the solution cannot be easily found. Numerical methods can be tried, and compared with the analytical solution.
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|Equations containing nonlinear operations
If there exist nonlinear functions of the highest-order derivative of the unknown function, the direct method discussed earlier cannot be used to find the first-order explicit differential equations in standard form. The algebraic equation solution process should be embedded in the differential equation description. Finally, the numerical solutions can be found. An example next will be used to demonstrate how to convert and solve such differential equations.
Example 4.11. Consider a more complicated initial value problem:
$$
\left(y^{\prime \prime}(t)\right)^{3}+3 y^{\prime \prime}(t) \sin y(t)+3 y^{\prime}(t) \sin y^{\prime \prime}(t)=\mathrm{e}^{-3 t}, \quad y(0)=1, y^{\prime}(0)=-1
$$
微分方程代写
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|EQUATIONS CONTAINING THE SQUARE OF THE HIGHEST-ORDER DERIVATIVE
假设未知函数的最高阶项以平方形式出现
[是(n)(吨)]2=F(吨,是(吨),是′(吨),…,是(n−1)(吨))
和初始值是(吨0),是′(吨0),…,是(n−1)(吨0)给出。和以前一样，状态变量X1(吨)=是(吨),X2(吨)=是′(吨),…,Xn(吨)=是(n−1)(吨)可以先选择，取最后一项的平方根，可以创建两组不同的一阶显式微分方程
$$
\left{\begin{aligned}
x_{1}^{\prime}(t) &=x_{2}(t) \
& \vdots \
x_{n-1}^{\prime}(t) &=x_{n}(t) \
x_{n}^{\prime}(t) &=\sqrt{f\left(t, y(t), y^{\prime}(t), \ldots, y^{(n-1)}(t)\right)}
\end{aligned}\right.
$$
and
$$
\left{\begin{aligned}
x_{1}^{\prime}(t) &=x_{2}(t) \
& \vdots \
x_{n-1}^{\prime}(t) &=x_{n}(t) \
x_{n}^{\prime}(t) &=-\sqrt{f\left(t, y(t), y^{\prime}(t), \ldots, y^{(n-1)}(t)\right)} .
\end{aligned}\right.
$$
The two state space models comprise the original differential equation. The initial values of the states are
$$
\boldsymbol{x}\left(t_{0}\right)=\left[y\left(t_{0}\right), y^{\prime}\left(t_{0}\right), \ldots, y^{(n-1)}\left(t_{0}\right)\right]^{\mathrm{T}}
$$
可以看出这两个微分方程组都可以直接用MATLAB求解器求解。因此，使用匿名函数或 MATLAB 函数，可以描述这两个方程组，以便对其进行数值求解。两种解都满足原微分方程。
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|EQUATIONS CONTAINING ODD POWERS
在实际应用中，甚至还有复杂形式的微分方程。例如，可能存在最高阶导数项的奇次幂：
[是n(吨)]2到+1=F(吨,X1(吨),X2(吨),…,Xn(吨)).
在实数域中，与平方对应物相比，此类微分方程的操作看起来更简单，因为根可能没有多重解的现象。可以像往常一样选择状态变量，例如，X1(吨)= 是(吨),X2(吨)=是′(吨),…,Xn(吨)=是(n−1)(吨). 最后，一阶显式微分方程可以写为
X′(吨)=[χ2(吨) ⋮ Xn−1(吨) F(吨,X1(吨),χ2(吨),…,Xn(吨))2到+1].
接下来的两个例子是作者根据给定的函数创建的是(吨)= 和−吨. 如果要以通常的方式找到解析解，则不容易找到解。可以尝试数值方法，并与解析解进行比较。
数学代写|微分方程代写DIFFERENTIAL EQUATION代考|EQUATIONS CONTAINING NONLINEAR OPERATIONS
如果存在未知函数的最高阶导数的非线性函数，则前面讨论的直接方法不能用于找到标准形式的一阶显式微分方程。代数方程求解过程应嵌入微分方程描述中。最后，可以找到数值解。接下来将使用一个示例来演示如何转换和求解此类微分方程。
例 4.11。考虑一个更复杂的初始值问题：
(是′′(吨))3+3是′′(吨)没有是(吨)+3是′(吨)没有是′′(吨)=和−3吨,是(0)=1,是′(0)=−1
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