I am curious how to approach solving the initial value problem: $\begin{cases} y'(t) = 5t - 3\sqrt{y} \\ y(0) = 2 \end{cases}$.
The equation isn't separable, and more generally it is not an exact equation. Nor does it seem to be readily convertible into an exact equation. I am interested in obtaining at least an implicit expression for $y$. Is it possible to use a Laplace transform to solve this nonlinear IVP? If not, what approach might one take?
With $v(t) = \sqrt{y(t)}$, the differential equation becomes $$ 2 v v' = 5 t - 3 v$$ which has the implicit solution (found with help of Maple) $$ (2 v + 5 t)^5 (v - t)^2 = c $$ To satisfy the initial condition, you need $c = 256 \sqrt{2}$. Thus $$ (2 \sqrt{y} + 5 t)^5 (\sqrt{y} - t)^2 = 256 \sqrt{2}$$