Question

已知齐次线性方程（x-1）y''-xy'+y=0的通解为Y=C1x+C2x^2,求非齐次线性方程（x-1）y''-xy'

Answer 1

（x-1）y''-xy'+y=（x-1）^2的齐次方程是（x-1）y''-xy'+y=0,

（x-1）y''-xy'+y=（x-1）^2解的结构就是（x-1）y''-xy'+y=0的通解加（x-1）y''-xy'+y=（x-1）^2的特解.

由于（x-1）^2不含e^x项,用待定系数法,设y*=ax^2+bx+c,把y*带入（x-1）y''-xy'+y=（x-1）^2,得到-ax^2+2ax-2a+c=x^2-2x+1,a=-1,b=0,c=-1

通解为Y=C1x+C2x^2-x^2-1=C1x+C2x^2-1