Mathematical Methods in the Physical Sciences · Problem 5.18

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · convolution-differential-equations

Problem

A mechanical or electrical system is described by the differential equation

y'' + \omega^2 y = f(t)

. Find

y

f(t) = \begin{cases} 1, & 0 < t < a, \\ 0, & \text{otherwise,} \end{cases} \quad \text{and} \quad y_0 = y_0' = 0.

Hint: Use the convolution integral carefully. Consider

t < a

and

t > a

separately, remembering that

f(t) = 0

for

t > a

. Show that

y = \begin{cases} \dfrac{1}{\omega^2}(1 - \cos \omega t), & t < a, \\ \dfrac{1}{\omega^2}[\cos \omega(t-a) - \cos \omega t], & t > a. \end{cases}

Sketch the motion if

a = \frac{1}{3}T

where

T

is the period for free vibrations of the system; if

a = \frac{3}{2}T

; if

a = \frac{1}{10}T