Solving xy'(x) - y(x) = x^2 Exp[x] using the power series method

[Repetit]

I've tried solving the equation xy'(x) - y(x) = x^2 Exp[x] using the power series method. I assume that y has the form:

$$y = \sum_{n=0}^{\infty} a_n x^n$$

Inserting this in the diff. eq. gives:

$$\sum_{n=0}^{\infty} n a_n x^n - \sum_{n=0}^{\infty} a_n x^n = x^2 e^x$$

Now, in the other types of diff. eqations I've solved using this method I found a recursive formula for the coefficients, but I don't see how I can do that here? How would I got about solving this problem?

Thanks

 

[cristo]

You could try using the series for e^x on the RHS. It's just a suggestion, so I don't know whether it'll work, but give it a go!

 

[HallsofIvy]

Yes, if you are required to use the series method, expand e^x in Taylor's series and compare corresponding powers of x.

Of course, that's an "Euler type" equation so it should be easier to solve it directly, perhaps by converting to an equation with constant coefficients.

 

[Repetit]

My problem is that I don't see how to obtain a recusion formula for the coefficients. Even when trying to solve just the corresponding homogeneous equation $$x y'(x) - y(x) = 0$$ I end up with something like $$n a_n - a_n = 0$$ and I can't see what to do.

 

[HallsofIvy]

Obviously, $na_n- a_n= 0$ is true as long as $a_n= 0$ for any n other than 1. For n= 1, the equation becomes $a_1= a_1$ which is also true- $a_1$ becomes the undetermined constant in your general solution. What that tells you is that the general solution to xy'(x)- y(x)= 0 is y(x)= Cx.
Since $e^x= \Sigma_{n=0}^\infty\frac{1}{n!}x^n$, $x^2e^x= \Sigma_{n=0}^\infty\frac{1}{n!}x^{n+2}$. If you let i= n+2, then you get $x^2e^x= \Sigma_{i=2}^\infty\frac{1}{(i-2)!}x^i$ so you can compare "like powers".
You get $(0)a_0- a_0= 0$ so a_0= 0. $(1)a_1- a_1= 0$ which is always true- $a_1$ will be an "unknown constant".

For n> 1, $na_n- a_n= a_n(n-1)= \frac{1}{(n-2)!}$ so
[tex]a_n= \frac{1}{(n-2)!(n-1)}[/itex].

 

[Repetit]

Hmm, of course. Why did I not see that. I guess I was to focused on obtaining a recursion formula for the coefficients, because I thought I always had to do that. I see now that I don't. Thanks a lot for the help!