Find the first five terms of the power series solution to the differential equation:
$$y' = y(1-y)$$
Letting $y = a_0+a_1x+a_2x^2+a_3x^3+...$
It's evident that: $$y' = \frac{dy}{dx} = a_1+2a_2x+3a_3x^2+4a_4x^3+...$$
And from the differential equation: $$y' = y(1-y) = (a_0+a_1x+a_2x^2+a_3x^3+...)(1-a_0-a_1x-a_2x^2-a_3x^3-...)$$
However, I don't know how to handle the product of these two infinite series. The Cauchy product comes to mind, but I'm not sure how helpful it'll be here.
Is it possible to solve this differential equation using power series alone?