Non-homogeneous 2nd order diff eq involves power series

[diffeqnoob]

I just need a hint or something to see where I start. I'm at a loss for a beginning.

Consider the non-homogenous equation
y'' + xy' + y = x^2 +2x +1

Find the power series solution about x=0 of the equation and express your answer in the form:

y=a_0 y_1 + a_1 y_2 + y_p

where a_0 and a_1 are arbitrary constants. Give only the first three nonzero terms of each of the three seriesy_1,y_2, and y_p

Hint: Substitute y = \sum_{n=0}^{\infty}a_nx^{n} and equate coefficients to find a_n, n = 2,3,4,5

 

[StatusX]

Do you know how to take the derivative of y in that form? If so, plug it in, and then try to rearrange the expression so that you have an infinite linear combination of powers of x that is equal to 0. Since the powers of x are linearly independent, all these coefficients must equal to zero, which will give you an expression for a_n in terms of a_n-1 and maybe a_n-2. This is called the Frobenius method, if you want to look online for a better explanation.