# Solve ODE by Power Series - Singular Point

• Mar 11th 2013, 11:50 AM
sjmiller
Solve ODE by Power Series - Singular Point
Hi all,

So I am just wondering if anyone can confirm that the following is a solution of the DE. I'm not entirely sure how to differentiate a factorial so I am at crossroads for checking to see if it satisfies the DE. This was a question off an exam I wrote this morning, and I'm just wondering if that solution is indeed correct.

$\displaystyle xy'' + y' - x^2y = 0$

Solution = $\displaystyle 1 + \sum^{\infty}_{n=1}{\frac{x^{3n}}{(n^2)!}$

• Mar 11th 2013, 12:17 PM
peysy
Re: Solve ODE by Power Series - Singular Point
y=a+b*x+c*x^2+d*x^3+e*x^4...
y'=b+2c*x+3d*x^2+4e*x^3 and so on ...you equate multipliers in both sides of DE...
• Mar 11th 2013, 02:19 PM
Prove It
Re: Solve ODE by Power Series - Singular Point
Quote:

Originally Posted by sjmiller
Hi all,

So I am just wondering if anyone can confirm that the following is a solution of the DE. I'm not entirely sure how to differentiate a factorial so I am at crossroads for checking to see if it satisfies the DE. This was a question off an exam I wrote this morning, and I'm just wondering if that solution is indeed correct.

$\displaystyle xy'' + y' - x^2y = 0$

Solution = $\displaystyle 1 + \sum^{\infty}_{n=1}{\frac{x^{3n}}{(n^2)!}$