微分方程

普通应对策略

分离变量，降阶
如
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伯努利




②

一阶线性齐次微分方程

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一阶线性非齐次微分方程

可能的正向思路

不难发现，中间得到的,正好就是对应齐次微分方程的通解，因此可以先求齐次通解再用常数变易法求解。

二阶线性解的结构

① 若为齐次的线性无关特解，则为齐次方程的通解
②若为齐次方程的通解，为非齐次方程的一个特解，则为非齐次线性方程的通解
③叠加原理，若分别为与的两个特解，则为的一个特解

二阶齐次

已知一特解求通解

将y=uv的方式推广到二阶非齐次

显然这个思路是可以推到n阶的
不过用这样的方式，求解一个重点就是与中的求解,这实际上就是求对应齐次方程的两个特解
剩下来就是联立求解。实际题目中，往往会给出齐次方程的一个特解或是通解，只需求得即可。

已知二阶齐次方程的通解

不正好就是这个方程的两个特解吗？
看看上面的内容，对比一下，正好就可以匹配，剩下的部分就是

可解得

已知二阶齐次方程一特解

这儿
令


代入原方程



特解可以代换


令
原方程则变成了一个一阶非齐次线性方程

二阶常系数齐次

这是二阶齐次的特殊情况，但是很具有讨论意义，均为常数，不难猜测是很符合要求的
代入原方程

特征方程
①，得两特解，通解为
②,得一特解,剩余一根，代入，其中

又

因此通解即为
得到结论另外一个特解为
③，
欧拉公式
两特解


根据解的叠加原理

即通解
结论
对于特征方程

特征根	通解

:::info


有

:::

二阶常系数非齐次

求解非齐次特解

情况一

设，其中为多项式

对于特征方程
① 不为特征方程的根
则
因此可以认为为m次多项式，即可设
② 为特征方程的单根

因此可以认为为m+1次多项式，即可设
③ 为特征方程的重根

因此可以认为为m+2次多项式，即可设
随后使用待定系数法即可得到特解

情况二

根据欧拉公式

根据解的叠加原理
可以令

则求解这两个微分方程的特解可以得到原微分方程的特解


根据情况一中的计算

如果,则
此时可设
否则
此时可设
可得

如果,则
此时可设
否则
此时可设
可得

其中
再次使用欧拉公式

因为共轭关系,可以拆开该式子消除复数部分
即
在解该类型时,可假设,随后使用待定系数法求解
推广n阶,只需考虑的变化,是与作为特征方程的根的次数有关的

欧拉方程

令

使用记号表示对求导
则


原方程转为

此时即为关于t的n阶常系数非齐次微分方程