(a) The differential equation for a critically damped harmonic oscillator, expressed in units chosen so that $\omega_0^2=1$, is
$$
\ddot{x}+2 \dot{x}+x=0 .
$$
Solve this equation by matrix methods, introducing $v=\dot{x}$ as a new variable. Write down the solution for initial conditions $\binom{x}{v}=\binom{x_0}{0}$ and for $\binom{x}{v}=\binom{0}{v_0}$, and sketch phase portraits of these and other solutions. Show that $x=0$ or $v=0$ can occur at most once.
(b) One way of solving the above equation without having to contend with a repeated eigenvalue is first to solve $\bar{x}+2 \dot{x}+\left(1-\varepsilon^2\right) x=0$, which leads to a matrix with distinct real eigenvalues, then let $\varepsilon \rightarrow 0$. (Physically, this corresponds to using a slightly weaker spring.) Carry through the procedure, first finding solutions for initial conditions $\binom{x_0}{0}$ and $\binom{0}{v_0}$, then letting $\varepsilon \rightarrow 0$. Show what happens to the phase portraits as $\varepsilon>0$.
(c) Another alternative is first to solve $\vec{x}+2 \dot{x}+\left(1+\varepsilon^2\right) x=0$, which leads to a matrix with complex eigenvalues, then let $\varepsilon \rightarrow 0$. Do this, again showing what happens to the solutions for initial conditions $\binom{x_0}{0}$ and $\binom{0}{v_0}$, and to the phase portraits, as $\varepsilon \rightarrow 0$.