Exercise 30

The differential equation $$\frac{dy}{dt} = y - 4t + y^2 - 8yt + 16 t^2 + 4.$$ is not autonomous, separable, or linear; however, we can solve this equation with a change of variable.

1. Transform this equation into a new differential equation of the form $$\frac{du}{dt}= f(u)$$ by letting $u = y - 4t$.

2. Sketch the phase line for this new equation, $u' = f(u)$, and sketch several solutions.

3. Find the solutions of the original differential equation that correspond to the equilibrium solutions of $u' = f(u)$. Graph these solutions in $ty$-plane. Also, sketch the graphs of the solutions that you plotted in part (b).

4. Solve the differential new equation and use this information to solve the original differential equation.