Series solution for differential equation

[Sam2000009]

<OP warned about not using the homework template>

Obtain a series solution of the differential equation x(x − 1)y" + [5x − 1]y' + 4y = 0Do I start by solving it normally then getting a series for the solution or assume y=power series differentiate then add up the series?

I did the latter and got three different but (slightly similar) looking series and the problem asks to group them all into one series which I cannot do

 

[BvU]

I did the latter and got three different but (slightly similar) looking series and the problem asks to group them all into one series which I cannot do

You have a series for y, e.g. $y = \displaystyle \sum_{n=0}^\infty c_nx^n$ so that $$y' = \sum_{n=1}^\infty n c_nx^{n-1}\quad{\rm {and}}\quad y'' = \sum_{n=2}^\infty n (n-1) c_nx^{n-2} \ , $$ right ?

Now work out the coefficient of $\ x^n$ in x(x − 1)y" + [5x − 1]y' + 4y = 0

 

[vela]

The problem asks to group them all into one series which I cannot do.

What's stopping you?