求方程的通解x∧2y''-4xy'+6y=x

Question

求方程的通解x∧2y''-4xy'+6y=x

Answer 1

设x=e^t
dx/dt = x
dy/dx = dy/(xdt)=(dy/dt)/x
d^2 y/d x^2
=d((dy/dt)/x)/dx
= d((dy/dt)/x)/(xdt)
= (d^2 y / dt^2) / x^2 -(dy/dt)/x^2
因此
x dy/dx = dy/dt
x^2 d^2 y / dx^2 = d^2 y / dt^2 - dy/dt
d^2 y / dt^2 - 5dy / dt + 6y = e^t
设特解= ke^t
代入方程
k-5k+6k=1
得k = 1/2
特解= 1/2 e^t