Series solution of first order ODE

[soverylost]

Homework Statement

Find two non-zero terms of the power series solution of
y' = 1 + y^2 ,y(0) = 0
by using series substitution y(x) = sum (k=0 to inf) [a][/k] *x^k

Homework Equations

The Attempt at a Solution

First take the derivative of the power series to get
y' = sum (k=0 to inf) (k+1)*[a][/k+1]*x^k

Plug y and y' into the original ODE, here is where my problem is.
I want the powers of x to match so that i can match the coefficients of the series and get a recursive relationship to find the non-zero terms. How do i deal with the y^2 term? How do I square a series and still get matching x-terms?

 

[LCKurtz]

Remember, you are only asked to find the first couple of terms. So you just multiply out the first few terms the long way. For example, to start multiplying out these two:

$$a_0 + a_1x + a_2x^2 + ...$$
$$b_0 +b_1x + b_2x^2 + ...$$

You would get:

$$a_0b_0 + (a_0b_1 + a_1b_0)x + ...$$

and do you see how to get all the x² terms if you need them? So do that method for multiplying the series for y by itself for as many terms as you need.