工数分析上
第五章 常微分方程
一阶微分方程
- 可分离变量的微分方程:\(\cfrac{dy}{dx}=f(x)g(y)\)
\[\int \cfrac{dy}{g(y)}=\int f(x)dx
\]
- 齐次方程:\(\cfrac{dy}{dx}=f(\cfrac{y}{x})\)
\[设u=\frac{y}{x}
\]
\[\cfrac{dy}{dx}=u+x\cfrac{du}{dx}~~=>\small{可分离变量}
\]
- \(\cfrac{dy}{dx}=f(\cfrac{ax+by+c}{a_1x+b_1y+c_1})\)
\[若\cfrac{a_1}{a}=\cfrac{b_1}{b}=\lambda
\]
\[设u=ax+by
\]
\[\cfrac{du}{dx}=a+bf(\frac{u+c}{\lambda u+c_1})~~~=>\small可分离变量
\]
\[若\cfrac{a_1}{a}\neq\frac{b_1}{b}
\]
\[\left\{ \begin{aligned} ax+by+c=0\\
a_1x+b_1y+c_1=0\end{aligned} \right.\]
\[\small求解得x=x_0,y=y_0
\]
\[设\xi=x-x_0,\eta=y-y_0,\cfrac{dy}{dx}=\cfrac{d\eta}{d\xi}
\]
\[\cfrac{d\eta}{d\xi}=f(\cfrac{a\xi+b\eta}{a_1\xi+b_1\eta})~~=>\small 齐次方程
\]
- 一阶线性微分方程:\(\cfrac{dy}{dx}+P(x)y=Q(x)\)
\[y=e^{-\int P(x)dx}[C+\int Q(x)e^{\int P(x)dx}dx]
\]
- 伯努利方程：\(\cfrac{dy}{dx}+P(x)y=Q(x)y^n(n\neq0,1)\)
\[\small两边同除y^n,得y^{-n}\cfrac{dy}{dx}+P(x)y^{1-n}=Q(x)
\]
\[\cfrac{1}{1-n}\cfrac{dy^{1-n}}{dx}+P(x)y^{1-n}=Q(x)
\]
\[设u=y^{1-n}
\]
\[\cfrac{du}{dx}+(1-n)P(x)u=(1-n)Q(x)~~=>\small一阶线性方程
\]
可降阶的高阶微分方程
- \(y^{(n)}=f(x)\)，多次积分
- \(y^{\prime\prime}=f(x,y^\prime)\)
\[设y^\prime=p(x),y^{\prime\prime}=p^\prime(x)
\]
\[p^\prime=f(x,p)~~=>\small 一阶微分方程
\]
- \(y^{\prime\prime}=f(y,y^\prime)\)
\[设y^\prime=p(y)
\]
\[p\frac{dp}{dy}=f(y,p)
\]