2018-52.md

course	course_year	question_number	tags	title	year
Methods	IB	52	IB 2018 Methods	Paper 2, Section II, A	2018

(a) Let $f(x)$ be a $2 \pi$-periodic function (i.e. $f(x)=f(x+2 \pi)$ for all $x$ ) defined on $[-\pi, \pi]$ by

$$f(x)=\left{\begin{array}{cl} x & x \in[0, \pi] \\ -x & x \in[-\pi, 0] \end{array}\right.$$

Find the Fourier series of $f(x)$ in the form

$$f(x)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty} a_{n} \cos (n x)+\sum_{n=1}^{\infty} b_{n} \sin (n x)$$

(b) Find the general solution to

$$y^{\prime \prime}+2 y^{\prime}+y=f(x)$$

where $f(x)$ is as given in part (a) and $y(x)$ is $2 \pi$-periodic.