any 1 can help me on solving the following question plz.
Use Frobenius method to solve the following differential equation
X2Y'' +X(2-X)Y'-2Y= 0
This is how I'd start it and I'm taking this right out of Rainville and Bedient on the chapter on power series solutions:
$\displaystyle 2xy''+x(2-x)y'-2y=0$
Now convert it to the standard form $\displaystyle y''+p(x)y'+q(x)y=0$:
$\displaystyle y''+\frac{x(2-x)}{2x} y'-\frac{1}{x}y=0$
with $\displaystyle p(x)=1-x/2$ and $\displaystyle q(x)=1/x$. That means $\displaystyle p_0=0$ and $\displaystyle q_0=0$ leaving for the indicial equation $\displaystyle c^2-c=0$ giving roots of 0 and 1. So I'd next go to the section dealing with difference of roots a positive integer by first substituting $\displaystyle y(x)=\sum_{n=0}^{\infty} a_n x^{n+c}$ into the original differential equation and continuing.
Razem, I had some problems going further with this as I've been away from it for some time but I do know the route if I had to solve it: Open the book to the first page of that chapter on power series, start reading, do all of the examples, do a few problems in each section, let it simmer over several days, maybe I don't know about 5-10 problems in all. Eventually, I'd get to the point where I could then go back to this problem and work it. That's really in my opinion how to successfully approach a problem you can't solve in math and elsewhere: put it on the back-burner and work some simpler ones first and then "scale-up" .