1. Consider the differential equation \(\dot{y} = t^2 + 2\cos(t)\).
   1. What is the general solution to this differential equation?
   2. What is the particular solution \(y = f(t)\) to this differential equation corresponding to the initial condition \[f\left(\frac{\pi}{2}\right) = \frac{\pi^3}{24}\,?\]
   3. An adult is riding their tricycle along the straight road that passes by their house. Their velocity \(\dot y\) (mph) can be modelled by the differential equation \({\dot{y} = t^2 + 2\cos(t)}\) as a function of \(t\) hours since they left their house. If the tricyclist was \(\frac{\pi^3}{24}\) miles from their house \(\frac{\pi}{2}\) hours after they left, how far will the tricyclist be from their home in \(2\pi\) hours?
2. Consider the differential equation \( \ddot{y} +t(4+3t) = 2+2\dot{y}+3y\,. \)
   1. Show that \(y = t^2 + \mathrm{e}^{3t}\) is a particular solution.
   2. Show that \(y = t^2 - 7\mathrm{e}^{-t}\) is also a particular solution.
   3. Show that \(y = t^2 + C_1\mathrm{e}^{-t} + C_2\mathrm{e}^{3t}\) is the general solution.
   4. What is the particular solution \(y = f(t)\) to this differential equation corresponding to initial conditions \(f(0) = -30\) and \({f(\ln(2)) = \big(\!\ln(2)\big)^2\,?}\)
3. Show that the logistic curve \[ y = \frac{M}{1+C_1\mathrm{e}^{-kMt}} \] with parameters \(M\) and \(k\) is the general solution to the differential equation \({\dot{y} = k\left(\!M-y\right)y\,.}\)