Exercise 8

In this problem, we test further what it means for a function to be a solution to a given differential equation.

1. Consider the differential equation $$\begin{gathered} \frac{dy}{dt} = y - t \end{gathered}$$. Determine whether the following functions are solutions to the given differential equation.

   1. $y(t) = t + 1 + 2e^t$

   2. $y(t) = t + 1$

   3. $y(t) = t + 2$

2. When you weigh bananas in a scale at the grocery store, the height $h$ of the bananas is described by the differential equation $$\begin{gathered} \frac{d^2h}{dt^2} = -kh \end{gathered}$$ where $k$ is the spring constant, a constant that depends on the properties of the spring in the scale. After you put the bananas in the scale, you (cleverly) observe that the height of the bananas is given by $h(t) = 4\sin(3t)$. What is the value of the spring constant?