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 series$$y_1$$,$$y_2$$, 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.