I've been revising Power series recently and their application when it comes to solving linear differential equations, but in this question I'm not sure what to do when it's a non linear function. I tried doing the same steps I do normally but these end up a mess. Any help would be appreciated.
Consider the differential equation $y'-y^{2}=0$ with initial condition $y(0)=c$. Notice that this is a non-linear equation. Suppose that $f(x)= \sum_{n=0}^{\infty} a_{n}x^{n}$ is a power-series solution.
$(i)$ Find $ a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$
$(ii)$ Guess the correct formula for $a_{n}$
$(iii)$ Assuming your guess in (b) is correct, find the radius of convergence of your powersolution.
You still equate coefficients, as always. The $y'$ term looks like
$$\sum_{n=0}^{\infty} (n+1) a_{n+1} x^n$$
The $y^2$ term is guided by the following:
$$y^2(x) = \sum_{n=0}^{\infty} c_n x^n$$
where
$$c_n = \sum_{m=0}^n a_m \, a_{n-m}$$
So, equating coefficients of $x^0$, we get $a_1-a_0^2=0$. Equating coeffs of $x^1$, we get
$$2 a_2=2 a_0 a_1$$
and so on.