1. The problem is to solve the differential equation $$\frac{dy}{dt} = (ty)^2$$. 2. This is a separable differential equation because the right side can be written as a product of a function of $t$ and a function of $y$. 3. Rewrite the equation as $$\frac{dy}{dt} = t^2 y^2$$. 4. Separate variables: $$\frac{dy}{y^2} = t^2 dt$$. 5. Integrate both sides: $$\int y^{-2} dy = \int t^2 dt$$ 6. The integral of $y^{-2}$ is $$-y^{-1} + C_1$$ and the integral of $t^2$ is $$\frac{t^3}{3} + C_2$$. 7. Combine constants into one constant $C$: $$-\frac{1}{y} = \frac{t^3}{3} + C$$. 8. Solve for $y$: $$y = -\frac{1}{\frac{t^3}{3} + C}$$. 9. This is the implicit general solution to the differential equation. 10. If an initial condition is given, substitute $t$ and $y$ values to find $C$. Final answer: $$y = -\frac{1}{\frac{t^3}{3} + C}$$