Q7. Solve the initial value problem
$x'(t) = Ax(t) + \begin{pmatrix} e^{-t} \\ -e^t \end{pmatrix}$, $x(0) = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$
where $x \in C^1([0, T]; \mathbb{R}^2)$ and
$A = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$.
(3 marks)
Q8. Prove the second shift theorem for Laplace transforms. That is show that for a function $f : [0, \infty) \to \mathbb{R}$ of
exponential order and piecewise continuous we have
$\mathcal{L}[t \to H(t - c)f(t - c)](s) = e^{-cs}\mathcal{L}[f](s)$.
for all $c > 0$ constant such that the Laplace transforms in the above relation make sense.
(3 marks)
Q9. Consider the BVP given by $y''(t) + 36ay(t) = 0$, $y(0) = 1$ and $y(\pi/6) = 1$.
Find all constants $a \in \mathbb{R}$ such that the BVP has a unique solution.
(2 marks)
