2011-13.md

course	course_year	question_number	tags	title	year
Complex Methods	IB	13	IB 2011 Complex Methods	Paper 4, Section II, D	2011

State and prove the convolution theorem for Laplace transforms.

Use Laplace transforms to solve

$$2 f^{\prime}(t)-\int_{0}^{t}(t-\tau)^{2} f(\tau) d \tau=4 t H(t)$$

with $f(0)=0$, where $H(t)$ is the Heaviside function. You may assume that the Laplace transform, $\widehat{f}(s)$, of $f(t)$ exists for Re $s$ sufficiently large.