chap 6
10 常微分方程初步¶
10.1 微分方程的基本概念¶
常微分方程 偏微分方程¶
通解 奇解¶
解的形式¶
- 显式给出:y = f(x).
- 隐式通解(通积分): \Phi(x, C_1,C_2, \cdots, C_n) = 0
10.2 一阶微分方程的初等解法¶
可分离类型¶
{d y\over dx} = f(x)g(y)\\
\Rightarrow {dy\over g(y)} = f(x)dx.
几种可化为可分离类型¶
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{dy \over dx} = f({y\over x}).\\ 令u = {y\over x} \Rightarrow {dy \over dx} = u + x\cdot {du\over dx}
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{dy\over dx} = {a_1x + b_1 y + c_1 \over a_2 x +b_2 y + c_2}\\ 令\begin{cases} u = a_1x + b_1 y + c_1\\ v = a_2 x +b_2 y + c_2 \end{cases}\\ 解一个二元一次方程组得到dx, dy.\\ 可能还要用上一种方法再换一次元.
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观察法: u = xy, u = ax + by, etc.
10.3 一阶线性微分方程¶
解法的推导¶
考虑一阶线性常微分方程的形式,\\
y' + P(x)y = Q(x)\\
我们给等式两端乘以一个不为零的\bold{积分因子}\mu(x) \\
\mu(x)y' + \mu(x)P(x)y = \mu(x)Q(x)\quad (*)\\
另外的,我们可以设\\
\mu y' + \mu P y = (\mu y)'\quad(**)\\
\iff \mu P y = \mu' y \iff \mu ' = \mu P\\
\Rightarrow \mu(x) = e^{\int P(x)dx}\\
由(*)(**)\\
(\mu y)' = \mu Q\\
由此得到通积分,\\
\mu(x) y = \int \mu(x) Q(x) dx +C\\
\Rightarrow y =(\mu (x))^{-1} [\int \mu(x) Q(x) dx +C],\\
式中\mu(x) = e^{\int P(x)dx}.
Bournelli方程¶
对于一阶常微分方程的更广泛形式,\\
y' + Py = Qy^\alpha,\\
当y \ne 0时,两边同乘y^{-\alpha},\\
y^{-\alpha}y' + Py^{1-\alpha} = Q.\\
\iff {1\over 1- \alpha}\cdot {d(y^{1-\alpha})\over dx} + Py^{1-\alpha} = Q\\
令u = y^{1-\alpha},\\
\Rightarrow u' + (1 - \alpha)Pu = (1-\alpha ) Q.\\
于是方程转化为关于u(x) 的线性微分方程.
10.4 全微分方程与积分因子¶
10.6 高阶微分方程¶
10.6.2 二阶线性方程¶
线性相关性的判断¶
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线性无关 $$ \begin{align} (1)& Wronski \to \exist x_0\in [a,b], s.t.\quad W(x_0) \ne 0\ \Rightarrow (2)& Liouville \to \forall x \in [a,b], W(x) \ne 0. \end{align} $$
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线性相关 $$ \begin{align} (1)& Liouville \to \forall x \in [a,b], W(x) = 0.\ (2)& Wronski \to \exist x_0\in [a,b], s.t.\quad W(x_0) = 0.\
\end{align} $$
10.7 级数与微分方程¶
- 例子: 1 + x + x^2 +\cdots = {1\over 1- x}
观察
法一: $$ S(x) = 1 + x + x^2 + x^3 + \cdots\ xS(x) = x +x^2 +x^3 + \cdots = S(x) - 1.\ \Rightarrow xS(x) = S(x) -1.\ $$ 法二: $$ S'(x) = 1+ 2x + 3x^2 +\cdots\ xS'(x) = x + 2x^2 + 3x^3 + \cdots\ (1-x)S'(x) = S(x)\ \iff y'(1-x) = y $$
例10.7.3 求级数\displaystyle 1+ \sum {(2n-1)!!\over 2^n\cdot n!}x^n的和函数S(x).
S'(x) =\displaystyle \sum_{n=1}^\infin {(2n-1)!!\over 2^n\cdot (n-1)!}x^{n-1} = {1\over 2} + \sum_{n=2}^\infin {(2n-1)!!\over 2^n\cdot (n-1)!}x^{n-1} = {1\over 2} + \sum_{n=1}^\infin {(2n+1)!!\over 2^{n+1}\cdot (n)!}x^{n}\\ xS'(x) = \sum {(2n-1)!!\over 2^n\cdot (n-1)!}x^{n} = \sum {(2n-1)!!\cdot n\over 2^n\cdot (n-1)!\cdot n}x^{n}\\ S'(x) - xS'(x) = {1\over 2} + \sum_{n=1}^\infin {(2n-1)!!\over 2^nn!}({2n+1\over 2} -n)x^n = {1\over 2} +{1\over 2}\sum {(2n-1)!!\over 2^n\cdot n!}x^n = {1\over 2 }S(x)\\ \Rightarrow y'- xy' = {1\over 2 } y\quad y|_{x= 0} = 1 \quad y'|_{x=0} = {1\over 2}.