QUESTION 3
LAPLACE TRANSFORMS
By applying Laplace transforms, solve the following initial value problems. You may use the
tables included in the resource sheet at the end of the examination paper, but in each instance you
must explain your working.
$\frac{dy}{dt}$ + 2y = $e^{2t}$, y(0) = 2.
(a)
(5 marks)
(b)
$\frac{d^2y}{dt^2}$ - 2$\frac{dy}{dt}$ = $\delta(t - 1) + \delta(t - 3)$, y(0) = 0, y'(0) = 1, where $\delta(t)$ is the Dirac delta
function.
(5 marks)
(c)
$\frac{d^2y}{dt^2}$ - 4y = f(t), y(0) = 0, y'(0) = 1, where
f(t) = \begin{cases} t^2, & 0 \le t < 1, \
0, & t \ge \pi. \end{cases}
(5 marks)
