Finding solution of nonlinear differential system

Question

$$x'=2x+y^2$$ $$y'=y$$

How to find a solution of above system if $x>0$?

I found a solution of second equation, $y=y_0e^t$, but don't know how to use this to solve the system. Thanks.

Answer 1

$$x'-2x=y^2$$ $$x'-2x=(y_0)^2e^{2t}$$ the characteristic equation for homogeneous D.E $$m-2=0$$ $$m=2$$ the complementary solution of $x$ $$x_c=C_1e^{2t}$$ so the particular solution $$x_p=Ate^{2t}$$ and then find the value of $A$