Problem W1.3 (complex-valued ODE's). Consider the general model linear second
order problem given by the forced harmonic oscillator IVP
$\ddot{u} - 2\mu \dot{u} + \lambda^2 u = g(x)$, $u(0) = u^0$, and $\dot{u}(0) = \dot{u}^0$,
(W1.3.1)
with $u \in \mathbb{R}$ and parameters $\lambda, \mu \in \mathbb{R}$, $\mu^2 \le \lambda^2$ and $g$ a real-valued continuous function
satisfying
$|g(x)| \le \frac{c_0}{x}$, for $x > 0$.
(W1.3.2)
(a) By introducing an appropriate auxiliary variable $v$, as a function of $u$ and $\dot{u}$,
show that problem (W1.3.1) is equivalent to the $\mathbb{R}^2$-valued first order problem
$\begin{bmatrix} \dot{u} \\ \dot{v} \end{bmatrix} = \begin{bmatrix} \mu & \sqrt{\lambda^2 - \mu^2} \\ \frac{-\lambda^2 + \mu^2}{\sqrt{\lambda^2 - \mu^2}} & \mu \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix} + \begin{bmatrix} 0 \\ g(x)/\sqrt{\lambda^2 - \mu^2} \end{bmatrix}$, and $\begin{bmatrix} u(0) \\ v(0) \end{bmatrix} = \begin{bmatrix} u^0 \\ \dot{u}^0 \end{bmatrix}$.
(W1.3.3)
