ODE常微分方程总结
教材：DLI-BSc-Mathematics-Documents/Ordinary Differential Equatrions.pdf at main · JavaZeroo/DLI-BSc-Mathematics-Documents (github.com)
第一章
Explicit First Order Equations
这种形式的 \[y' = f(x, y)\] 称为 'Explicit First Order Equations' 。
\(y'=f(x)\)
1. Equations with Separated Variables
\[y' = f(x)g(y)\]
这种可以直接变成 \(\frac{dy}{g(y)}=f(x)dx\)
积分完后 \[\int\frac{dy}{g(y)}=\int f(x)dx\] IVP: \(y(\xi)=\eta\) \[\int^y_{\eta}\frac{dy}{g(y)}=\int^x_{\xi} f(x)dx\] 这里需要注意的是如果\(g(y(\xi))=g(\eta)=0\) 那么，直接就有 \(y'=0\) 因此\(y=\eta\) ;
2. 普通的替换
\[y'=f(ax+by+c)\]
这种情况用\(u(x) = ax+by+c\) 去替换掉变量\(x\) 。原理是\(u'=a+by'(x)=a+bf(u)\)。很关键的一点是，\(u\) 和\(y\) 都是一次的。求导后刚好是线性关系。
因此 最后得出的\(u(x)\)后可以直接利用\(u(x)=ax+by+c\)得到\(y\)。
3. 普通的Homogeneous Differential Equation
\[y'=f\left(\frac{y}{x}\right)\]
同样的道理用\(u(x) = \frac{y(x)}{x}, (x\neq0)\)替换掉变量\(y\)。有\(y'=u+xu'=f(\frac{y}{x})\) ，可得\(u'=\frac{f(u)-u}{x}\)
因此 最后得出的\(u(x)\)后可以直接利用\(u(x)=\frac{y(x)}{x}\)得到\(y\)。
4. 高级的 Homogeneous Differential Equation
\[y'=f\left(\frac{ax+by+c}{\alpha x+\beta y+ \gamma}\right)\]
这个的核心思想是转换成“普通的Homogeneous Differential Equation”。
首先分析行列式 \[\left | \begin{matrix}a &b \\\alpha &\beta \\\end{matrix} \right | \]
行列式为零时，说明\(a=\lambda_a \alpha, b=\lambda_b \beta\) 此时 可以直接转换成“普通的Homogeneous Differential Equation”
行列式不为零的时候，说明方程组有唯一解。
首先解方程组 \[\left\{\begin{align} ax+by+c&=0\\ \alpha x+\beta y+ \gamma&=0\\\end{align}\right.\] 可以解出一组\((x_0, y_0)\) 利用这组解将“高级的 Homogeneous Differential Equation”转换成“普通的Homogeneous Differential Equation”。
原理是新建坐标系得\(\bar{x}:=x-x_0, \bar{y}:=y-y_0\) 那么在这个坐标系下面原方程就变成了\(\bar{y}(\bar{x}):=y(\bar{x}+x_0)-y_0\)。对这个方程求导可以将\(y\)消掉，将原问题变\(\bar{y}\)和\(x\)的关系。这样做的目的就是将“高级的 Homogeneous Differential Equation”转换成“普通的Homogeneous Differential Equation”。
对\(\bar{y}\)求导后可以发现（利用方程\(\bar{y}(\bar{x}):=y(\bar{x}+x_0)-y_0, ax_0+by_0+c=0, \alpha x_0+\beta y_0+ \gamma=0\)） \[\frac{\bar{y}(x)}{d\bar{x}} = y'(\bar{x}+x_0)=f\left(\frac{a\bar{x}+b\bar{y}(\bar{x})}{\alpha\bar{x}+\beta\bar{y}(\bar{x})}\right)\] 这里就可以像刚才的普通的Homogeneous Differential Equation一样去做了。值得注意的是，这样解出来的\(\bar{y}\)需要转换成原来的\(y\)。这里可以用 \[y(x):=y_0+\bar{y}(x-x_0)\] 得到最后的\(y\)。
The Linear Differential Equation
这种形式的 \[y' + g(x)y=h(x)\] 称为 'The Linear Differential Equation' 。
这时有两种情况：\(h(x)=0\)和\(h(x)\neq0\)， 分别称为"homogeneous" 和"nonhomogeneous"
事实上当\(h(x)=0\)也就是"homogeneous"时，就是上面的"Explicit First Order Equations"，这里就不在赘述了。
对于\(h(x)\neq0\)的情况，也就是"nonhomogeneous"时，我们需要用到"Method of variation of constants"。
Method of variation of constants：这个方法首先计算出齐次的时候的通解。对于方程\(y'+g(x)y=h(x)\)他的齐次方程的通解是通过解\(y'+g(x)y=0\)，可得 \[y=C\cdot e^{-\int g(x)dx}\] 此时我们将常数\(C\)作为一个与\(x\)的函数\(C(x)\)。这个时候，只需要解出一个\(C(x)\)就是非齐次情况下的答案。
那么现在的问题就变成了，如何得到一个\(C(x)\)使得\(y' + g(x)y=h(x)\)？
首先先计算\(C'(x)\) \[\begin{align}y &=C(x)\cdot e^{-\int g(x)dx}\\y'&=\left( C(x) \right)'\cdot e^{-\int g(x)}+C(x)\cdot \left( e^{-\int g(x)dx }\right)' \\y'&=C'(x)\cdot e^{-\int g(x)}+C(x)\cdot \left( e^{-\int g(x)dx }\cdot\left(-\int g(x)\right)'\right) \\y'&=C'(x)\cdot e^{-\int g(x)}+C(x)\cdot \left( e^{-\int g(x)dx }\cdot -g(x)\right) \\&\Downarrow \\y'&=C'\cdot e^{-\int g(x)}-gC\cdot e^{-\int g(x)dx}\end{align}\] 然后代入\(y' + g(x)y=h(x)\) \[\begin{align}L_y \equiv y' + g(x)y&=h(x)\\L_y&=C'\cdot e^{-\int g(x)dx}-gC\cdot e^{-\int g(x)dx}+gC\cdot e^{-\int g(x)dx}\\L_y&=C'\cdot e^{-\int g(x)dx}=h(x) \\&\Downarrow \\C'(x)&=h(x)\cdot e^{\int g(x)dx} \\&\Downarrow \\C(x)&=\int h(x)\cdot e^{\int g(x)dx}dx + C_0\end{align}\] 现在我们可以知道，当\(y=C(x)\cdot e^{-\int g(x)dx}\)且\(C(x)=\int h(x)\cdot e^{\int g(x)dx} + C_0\)时。，有\(y' + g(x)y=h(x)\)。这里有个二级结论
If \(y, \bar{y}\), yare two solutions to the nonhomogeneous equation \(L_y = h\), then \(L(y - y) = L_y - L_{\bar{y}} = 0, i.e., z(x) = y - \bar{y}\) is a solution of the homogeneous equation \(L_y = 0\). Thus all solutions \(y(x)\) of the nonhomogeneous equation can be written in the form \[y(x)=\bar{y}+z(x)\]
这里面\(z(x)\)就是刚刚的\(y=C\cdot e^{-\int g(x)dx}\)，\(\bar{y}\)就是\(C(x)=\int h(x)\cdot e^{\int g(x)dx} + C_0\)和\(y=C\cdot e^{-\int g(x)dx}\)的结合中的\(y\)，也就是\(\bar{y}=\left(\int h(x)\cdot e^{\int g(x)dx}dx + C_0\right)\cdot e^{-\int g(x)dx}\)
Bernoulli's Equation
这种形式的 \[y'+g(x)y+h(x)y^{\alpha}=0.\alpha \neq 1\] 非常的简单，只需要把\(y^{\alpha}\)解决了就可以了。等式去除\(y^{\alpha}\)有 \[y'y^{-\alpha}+g(x)y^{(1-\alpha)}+h(x)=0\] 利用\(z=y^{(1-\alpha)} \implies z'=(1-\alpha)y^{-\alpha}\cdot y'\)替换原式得 \[\frac{1}{1-\alpha}z'+g(x)z+h(x)=0\] 现在，就变成了nonhomogeneous的"The Linear Differential Equation"。最后解出\(z\)，别忘了替换回\(y\)。
Exact differential equations
这种形式的 \[M(x,y)dx+N(x,y)dy=0,\\ \exists\ U(x, y)\ s.t.\ U_x(x,y)=M(x,y),U_y(x,y)=N(x,y)\]
\(xdx+ydy=0\) is an exact equation, and \(U(x,y)=1/2 (x^2+y^2 )\)is a potential function.
Integrating Factors
Integrating Factors是用来让非 Exact 变成Exact differential equations。
E.g. \(ydx + 2xdy = 0\) is not exact. However, it can easily be made an exact differential equation (in the domain \(x > 0\)) by multiplying the equation by \(1/\sqrt{x}\). The resulting differential equation \[\frac{y}{\sqrt{x}}dx+2\sqrt{x}dy=0\] is exact, and a potential function is given by \[F(x,y)=2y\sqrt{x}=0\ (x>0)\]
对于一个not excat differential equation我们需要找到一个Factor \(U(x,y)\) 使得\(U(x,y)\cdot M(x,y)dx+U(x,y)\cdot N(x,y)dy=0\)变成一个Exact differential equations。
这里有一个"Theorem on potential functions"保证可以找到\(U(x,y)\)
现在的问题就是，如何去找？首先令\(M' = U\cdot M,N'=U\cdot N\)如果\(F_x=M',F_y=N'\)则有\(M'_y=N'_x\)。利用这个关系可以知道
事实上就是\(F_{xy}=F_{yx}\) (Jacobian Matrix)
\[\begin{align}&(U\cdot M)_y=(U\cdot N)_x\\\implies &U_y\cdot M+U\cdot M_y=U_x\cdot N+U\cdot N_x \\\end{align}\] 此时需要考虑，Integrating Factors是只与\(x\)有关还是只与\(y\)有关（只需要选一个）
假如只与\(x\)有关则\(U_y=0\)则有
\[\begin{align}U_y\cdot M+U\cdot M_y&=U_x\cdot N+U\cdot N_x \\U\cdot M_y &= U' \cdot N+U \cdot N_x \\\frac{1}{U}U'&=\frac{M_y-N_x}{N} \\(\ln U)'&=\frac{M_y-N_x}{N} \\U&=e^{\int \frac{M_y-N_x}{N}dx}\end{align}\]
这里的答案不是\(U=C\cdot e^{\int \frac{M_y-N_x}{N}dx}\)的原因是，我们只需要找到一个\(U\)，因此你可以认为我们选择\(C=1\)作为答案。下面的情况同理。
假如只与\(y\)有关则\(U_x=0\)则有
\[\begin{align}U_y\cdot M+U\cdot M_y&=U_x\cdot N+U\cdot N_x \\U'\cdot M+U\cdot M_y &= U \cdot N_x \\\frac{1}{U}U'&=\frac{N_x-M_y}{M} \\(\ln U)'&=\frac{N_x-M_y}{M} \\U &=e^{\int \frac{N_x-M_y}{M}}\end{align}\]
Implicit First Order Differential Equations
这种形式的 \[F(x, y, y')=0\] 一般来说有两种解决办法。要么通过一些方法获得explicit differential equation，要么就用参数化。
在这里我们只讨论两种情况：
$F(x, y')=0 $, \(F(y, y')=0\)
\(y=f(x,y')\), \(x=f(y,y')\)
第一种情况
对于$F(x, y')=0 $我们使用参数化： \[\left\{\begin{array}{l}x=\phi(t) \\y'=\psi(t)\end{array}\right.\] 此时方程变为\(F(\phi(t), \psi(t))=0\)，同时我们有 $$ \[\begin{align}y'&=\frac{dy}{dx} \ \text{and} \ \phi'(t)=\frac{d\phi(t)}{dt} \\dy&=y'dx \ \ \text{and} \ d\phi(t)=\phi'(t)dt\\ y&=\int y'dx + C\\y&=\int \psi(t)d\phi(t) + C \\y&=\int \psi(t)\phi'(t)dt +C\end{align}\] \[最后得到：\] {\[\begin{array}{l}x=\phi(t) \\y=\int \psi(t)\phi'(t)dt +C\end{array}\]
. $$
第一种情况2
对于\(F(y,y')=0\)我们仍然参数化： \[\left\{\begin{array}{l}y=\phi(t) \\y'=\psi(t)\end{array}\right.\] 此时有\(F(\phi(t),\psi(t)=0\)，同时有： \[\begin{align}y'&=\frac{dy}{dx} \ \text{and} \ \phi'(t)=\frac{dy}{dt} \\dx&=\frac{dy}{\psi(t)} \ \text{and} \ dy=\phi'(t)dt \\dx&=\frac{\phi'(t)dt}{\psi(t)} \\\int dx&= \int \frac{\phi'(t)dt}{\psi(t)} \\x&=\int \frac{\phi'(t)dt}{\psi(t)}\end{align}\]
An Existence and Uniqueness Theorem
中文名是存在唯一性定理。首先我们介绍Lipschitz condition:
We consider the following initial value problem \[y' = f(x,y),\ \text{for}\ \xi \leq x\leq \xi+a,\ y(\xi)=\eta\] The main assumptions in the following theorem are that \(f\) is continuous in the strip \(S=J\times\mathbb{R}\) with \(J=[\xi,\xi+a]\) and satisfies a Lipschitz condition with respect to \(y\) in \(S\) \[|f(x,y)-f(x, \bar{y})|\leq L|y-\bar{y}|\] No restrictions are placed on the value of the Lipschitz constant \(L\geq 0\)
然后引出存在唯一性定理：
Let \(f \in C(S)\) satisfy the Lipschitz condition. Then the IVP has exactly one solution \(y(x)\). The solution exists in the interval \(J: \xi\leq x \leq\xi +a\)
The extension of solutions.
以下三个定理是用于 The extension of solutions.
1️⃣Local Lipschitz condition. The function \(f(x,y)\) is said to satisfy a local Lipschitz condition with respect to \(y\) in \(D\subset R^2\) if for every \((x_0,y_0 )\in D\) there exists a neighborhood \(U=U(x_0,y_0 )\) and an \(L=L(x_0,y_0 )\) such that in \(U\cup D\) the function \(f\) satisfies the Lipschitz condition \(|f(x,y)-f(x,\bar{y})|\leq L|y-\bar{y}|\).
注意！我们一般通过连续性来判断Local Lipschitz condition
If \(D\) is open and if \(f \in C(D)\) has a continuous derivative \(f_y\) in \(D\), then f satisfies a local Lipschitz condition in this set.
2️⃣Theorem on local solvability If \(D\) is open and \(f\in C(D)\) satisfies a local Lipschitz condition in \(D\), then the IVP is locally uniquely solvable for$ (x_0,y_0 )∈D$; i.e., there is a neighborhood \(I\) of$ x_0$ such that exactly one solution exists in \(I\).
3️⃣Theorem on the extension of solutions Let \(f \in C (D)\) satisfy a local Lipschitz condition with respect to \(y\) in \(D\). Then for every \((x_0, y_0)\in D\) the initial value problem \(y' = f (x, y), y(x_0) = y_0\) has a solution that can be extended to the left and to the right comes arbitrarily close to the boundary of \(D\).
最后我们有：
The Peano existence theorem. If \(f(x,y)\) is continuous in a domain \(D\) and \((\xi,\eta)\) is a point in \(D\), then at least one solution of the differential equation \(y′=f(x,y)\) goes through \((\xi,\eta)\). Every solution can be extended to the left and to the right up to the boundary of \(D\).
Linear System
这里开始就是在讨论，常微分方程组。
Systems of n Linear Differential Equations
给出常微分方程组的形式： $$ \[\begin{align}y_1' &= a_{11}(t)y_1+\cdots+ a_{1n}(t)y_n+b_1(t) \\&\ \ \vdots \\y_1' &= a_{11}(t)y_1+\cdots+a_{1n}(t)y_n+b_1(t) \\\end{align}\] \[或者：\] '=A(t)+(t)\ \ A(t)=(a_{ij}(t)), (t)=(b_1(t), , b_n(t))^ $$ 值得一提的是，存在唯一性定理在常微分方程组也同样适用。
Homogeneous Linear Systems
对于齐次的形式，常微分方程组就变成了： \[\mathbf{y}'=A(x)\mathbf{y}\] 此时，根据存在唯一性定理有： \[\exist \text{ exactly one solution } \mathbf{y}=\mathbf{y}(t;\tau,\boldsymbol{\eta})\ \forall \tau \in J, \boldsymbol{\eta}\in \R^n\text{ or }\C^n\] 当然，齐次常微分方程组有一些重要的性质：
$ $ in $ J$ is a solution of the homogeneous linear systems.
There exist \(n\) linearly independent solutions \(_1,\dots,\mathbf{y}_n\). Every such set of \(n\) linearly independent solutions is called a fundamental system of solutions. If \(\mathbf{y}_1,\dots,\mathbf{y}_n\) is a fundamental system, then every solution \(\mathbf{y}\) can be written in a unique way as a linear combination \(\mathbf{y}=C_1 \mathbf{y}_1+\dots+C_n \mathbf{y}_n\).
A system of \(n\) solutions \(\mathbf{y}_1,…,\mathbf{y}_n\) can be assembled into an \(n\times n\) solution matrix \(\Phi(x)=(\mathbf{y}_1,\dots,\mathbf{y}_n )\). If \(n\) solutions \(\mathbf{y}_1,\dots,\mathbf{y}_n\) are linearly independent, then \(\Phi(x)\) is a system of \(n\) solutions \(\mathbf{y}_1,\dots,\mathbf{y}_n\) can be assembled into an \(n\times n\) solution matrix \(\Phi(x)=(\mathbf{y}_1,\dots,\mathbf{y}_n )\). If \(n\) solutions \(\mathbf{y}_1,\dots,\mathbf{y}_n\) are linearly independent, then \(\Phi(x)\)is a Fundamental Matrix.
The Wronskian
现在讨论一下，齐次常微分方程的解，是线性无关还是线性相关。
The Wronskian. If \(\Phi(x)=(\mathbf{y}_1,\dots,\mathbf{y}_n )\) is a solution matrix of \(\mathbf{y}^′=A(x)\mathbf{y}\), then its determinant \(W(x)=|\Phi(x)|\)is called the Wronskian determinant.
Theorem If \(\mathbf{y}_1,\dots,\mathbf{y}_n\) are linearly dependent in \(J\), then the Wronskian \(W(x)\equiv0\).
Theorem If \(\mathbf{y}_1,…,\mathbf{y}_n\) is a fundamental system of equation \(\mathbf{y}'=A(x)\mathbf{y}\), then the Wronskian \(W(x)\neq0\) in \(J\).
Theorem. There exists a fundamental system of solutions for equation \(\mathbf{y}'=A(x)\mathbf{y}\).
因此，我们先求出\(n\)个解，然后再去判断这\(n\)个解的Fundamental Matrix的行列式，也就是The Wronskian，是否为零。
Inhomogeneous Systems
对于非齐次的常微分方程组，就是最开始样子： \[\mathbf{y}'=A(t)\mathbf{y}+\mathbf{b}(t)\] 下面这个定理类似线性代数，非齐次方程组的通解是，齐次方程组的通解+非齐次方程组的特解：
Theorem. Let \(\tilde{\mathbf{y}}(x)\)be a fixed solution of the inhomogeneous equation (1). If \(\mathbf{y}_0 (x)\) is an arbitrary solution of the homogeneous equation, then \(\mathbf{y}(x)=\tilde{\mathbf{y}}(x)+\mathbf{y}_0 (x)\)is a solution of the inhomogeneous equation, and all solutions of the inhomogeneous equation are obtained in this way.