The homogeneous equation

$$\ddot{y}+p(t) \dot{y}+q(t) y=0$$

has non-constant, non-singular coefficients $p(t)$ and $q(t)$. Two solutions of the equation, $y(t)=y_{1}(t)$ and $y(t)=y_{2}(t)$, are given. The solutions are known to be such that the determinant

$$W(t)=\left|\begin{array}{ll} y_{1} & y_{2} \\ \dot{y}_{1} & \dot{y}_{2} \end{array}\right|$$

is non-zero for all $t$. Define what is meant by linear dependence, and show that the two given solutions are linearly independent. Show also that

$$W(t) \propto \exp \left(-\int^{t} p(s) d s\right) .$$

In the corresponding inhomogeneous equation

$$\ddot{y}+p(t) \dot{y}+q(t) y=f(t)$$

the right-hand side $f(t)$ is a prescribed forcing function. Construct a particular integral of this inhomogeneous equation in the form

$$y(t)=a_{1}(t) y_{1}(t)+a_{2}(t) y_{2}(t),$$

where the two functions $a_{i}(t)$ are to be determined such that

$$y_{1}(t) \dot{a}_{1}(t)+y_{2}(t) \dot{a}_{2}(t)=0$$

for all $t$. Express your result for the functions $a_{i}(t)$ in terms of integrals of the functions $f(t) y_{1}(t) / W(t)$ and $f(t) y_{2}(t) / W(t)$.

Consider the case in which $p(t)=0$ for all $t$ and $q(t)$ is a positive constant, $q=\omega^{2}$ say, and in which the forcing $f(t)=\sin (\omega t)$. Show that in this case $y_{1}(t)$ and $y_{2}(t)$ can be taken as $\cos (\omega t)$ and $\sin (\omega t)$ respectively. Evaluate $f(t) y_{1}(t) / W(t)$ and $f(t) y_{2}(t) / W(t)$ and show that, as $t \rightarrow \infty$, one of the $a_{i}(t)$ increases in magnitude like a power of $t$ to be determined.