Question

Please, kindly help me solve the following ODE Equations:
1.) Use the method of inverse differential operator to solve the following differential equation:
y"-2y'-3y=x^2e^2x

2.) Solve the given differential equation by means of a power series about the given point(x note). Find the recurrent relation and two linear independent solutions
(1-x)y"+xy'-y=0, (x note)=o

Answer 1

2.
\[y=\sum_{n=0}^{\infty}a_nx^n\\
y'=\sum_{n=1}^{\infty}na_nx^{n-1}=\sum_{n=0}^{\infty}(n+1)a_{n+1}x^n\\
y''=\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2}=\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n\]
Now plug this in and be careful when simplifying.