Question

求x^2y的二次导-3xy的导数-5y=x^2lnx的通解

Answer 1

此乃欧拉方程。令 x=e^t, 则

y'=(dy/dt)(dt/dx)=e^(-t)dy/dt=(1/x)dy/dt,

y''=(d/dx)[(1/x)dy/dt] = (-1/x^2)dy/dt+(1/x)(d^2y/dt^2)(dt/dx)

= (1/x^2)[d^2y/dt^2-dy/dt],

代入原微分方程，得 d^2y/dt^2-4dy/dt-...