What is the approach for solving a first order Riccati differential equation?

[laura_a]

Homework Statement

For the d.e

y' = x^2+y^2

Show that the solution with y(0)=0 has a vertical asymptote at some point x_0,
Then I have to try and find the upper and lower bounds for x_0

I'm not able to solve this for y because when I bring the y^2 to the LHS

The Attempt at a Solution

I'm trying to learn differential equations on my own though readings and I'm having trouble getting the hang of it... for the above question I tried a few things such as

y' - y^2 = x^2

I've tried a number of methods including multiplying throughout by x and I can't
find an equation whose differential is y' - y^2 OR xy' - y^2 because of the
negative sign? SO my main prob is I can't begin to solve it for y(0)=0 because I
don't know how to find y? Any help will be much appreciated!

 

[Defennder]

This looks like a Riccati differential equation.