Consider the second-order differential equation for $y(t)$ in $t \geqslant 0$

$$\ddot{y}+2 k \dot{y}+\left(k^{2}+\omega^{2}\right) y=f(t) .$$

(i) For $f(t)=0$, find the general solution $y_{1}(t)$ of $(*)$.

(ii) For $f(t)=\delta(t-a)$ with $a>0$, find the solution $y_{2}(t, a)$ of $(*)$ that satisfies $y=0$ and $\dot{y}=0$ at $t=0$.

(iii) For $f(t)=H(t-b)$ with $b>0$, find the solution $y_{3}(t, b)$ of $(*)$ that satisfies $y=0$ and $\dot{y}=0$ at $t=0 .$

(iv) Show that

$$y_{2}(t, b)=-\frac{\partial y_{3}}{\partial b}$$