**Example 1 ** Solve the equation $y''+3y'+2y=h(t)$; $y(0)=8$ and $y'(0)=0$, where
$h(t)=\begin{cases}
4t & 0\le t \lt 1 \\
4 & 1\leq t \lt 2 \\
0 & t\ge t
\end{cases}$.

First create the unit step function, and plot $h(t)$ along with the homogeneous solution:

Plot the solution to the forced equation:

Graph all three functions to see the affect of the discontiuous forcing function.

**Example 2 ** Determine the qualitative behavior of the solution to the equation $2y''+2y'+5y=h(t)$; $y(0)=10$ and $y'(0)=0$, where
$h(t)=\begin{cases}
10 & 5\le t \lt 10 \\
0 & \text{elsewhere}
\end{cases}$.

Plot the homogeneous solution and the forcing function:

Now plot the solution to the forced equation:

And plot all three together:

**Example 3:** done on paper

**Example 4: **Use a CAS to help solve $2y''+y'+30y=f(t)$ where $f$ is a constant force of 150 from $t=5$ to $t=8$.