Solving a Non-Linear ODE: What Method Should I Use?

[lkh1986]

Homework Statement

Solve $$y'=x^2+y^2$$ with initial condition y(0)=1.

Homework Equations

This is a first order ODE.

The Attempt at a Solution

I have tried separable variable, exact, and homogeneous and non-homogeneous, but none of them work. It's neither linear nor Bernoulli.

Any clue on what method have I missed or should I tried? Thanks.

 

[Metaleer]

It is a http://en.wikipedia.org/wiki/Riccati_equation" .

 

[lkh1986]

It is a http://en.wikipedia.org/wiki/Riccati_equation" .

Thanks. I will try using some suitable transformation to reduce it to a solvable linear DE. :)

 

[lkh1986]

" Without knowing at least one solution, there is absolutely no chance to find any solutions to such an equation."
http://www.sosmath.com/diffeq/first/riccati/riccati.html

Hm, any idea how can I guess the first solution y1?