Solve this Differential Equation:
$\displaystyle x\frac{d^2y}{dx^2}+(2-x)\frac{dy}{dx}-y=2cosx$
Thanks in advance..
First solve the homogeneous equation:
$\displaystyle xy''+(2-x)y'-y=0$
by power series. Try and obtain the expression $\displaystyle \sum_{n=0}^{\infty} a_n\left(n(n-1)+2n\right) x^{n-1}-\sum_{n=1}^{\infty} n a_{n-1} x^{n-1}$ which reduces to $\displaystyle a_0\left(\frac{e^x}{x}-\frac{1}{x}\right)$. Suppose $\displaystyle y_1$ is a solution of the homogeneous equation. Then can you use reduction of order by letting $\displaystyle y=v y_1$ to obtain the solution of the non-homogeneous equation?
If I use $\displaystyle y=\frac{1}{x}v$ and go through the steps, I get:
$\displaystyle v''-v'=2\cos(x)$
Try and get to that part, then let $\displaystyle w=v'$, integrate twice to obtain v. Do the same with $\displaystyle \frac{e^x}{x}$.