Question

微分方程求解x^2y''-xy'-3y=2x

微分方程求解x^2y''-xy'-3y=2x

Answer 1

首先求特解，取y*=cx带入得到
-cx -3cx = 2x, c=-1/2, y*=-x/2
考虑x^2 y'' -xy' -3y =0
考虑
(xy)' = y +xy'
(xy)'' = 2y' + xy''
x^2y'' -xy' -3y = x^2y'' +2xy' -3xy'-3y
=x(xy)'' -3(xy)' =0
令p=(xy)'
xdp/dx -3p =0, dp/p = 3dx/x
lnp = 3lnx +C
p = cx^3
(xy)' = cx^3
xy = c1x^4 +c2
y = c1x^3 +c2/x
所以最终解为c1x^3 +c2/x-x/2

Answer 2

该微分方程属于 y''+P(x)y'+Q(x)y=f(x) 类型的题。

该题的求解过程如下：