Chapter 4.4, Problem 30P

The position of a certain spring-mass system satisfies the initial value problem

$\begin{array}{ccc}y\text{'}\text{'}+\frac{1}{4}y\text{'}+2y=0,& y\left(0\right)=0,& y\text{'}\left(0\right)=2\end{array}$.

Find the solution of this initial value problem.

Plot $y$ versus $t$ and $y\text{'}$ versus $t$ on the same axes.

Draw the phase portrait for the dynamical system equivalent to $y\text{'}\text{'}+\frac{1}{4}y\text{'}+2y=0$. Include the trajectory corresponding to the initial conditions $y\left(0\right)=0,y\text{'}\left(0\right)=2$.