How to solve this differential equation dy/dx=x^2-sqrt(y)? Thanks.
I only have two ideas, neither one of which is going to be terribly good news. The first idea is to attempt a series solution. Before doing so, you might want to transform the equation thus: let $\displaystyle u=\sqrt{y},$ implying that $\displaystyle u^{2}=y.$ Then $\displaystyle 2uu'=y',$ and the DE becomes
$\displaystyle 2uu'=x^{2}-u.$
It might be easier to work with the series solution in this format than with the $\displaystyle y$ equation.
The second idea is to solve the DE numerically. You'll need an initial condition to pull that off.
What is the context for this DE? Does it arise from a physical problem?