# Name________________________ Student I.D.___________________ Math 2280-001 Quiz 12 ```Name________________________
Student I.D.___________________
Math 2280-001
Quiz 12
April 24, 2015
You may choose to do #1 (Laplace transform), or #2 (Fourier series). Indicate clearly which problem
1) Find a formula for solutions to the forced oscillation problem below, as convolution integrals involving
the forcing function f and the &quot;weight function&quot;. Hint: when you take the Laplace transform of the
differential equation and solve for X s if will be of the form X s = F s G s .
x## t C 2 x# t C 10 x = f t
x 0 =0
x# 0 = 0
(10 points)
2) Find the Fourier series for the square wave of period P = 2 L = 4 and amplitude 3. In other words, for
the period 4 extension of
K3 K2 ! t ! 0
f t =
.
3 0!t!2
You may do this directly from the definition
N
N
a0
p
p
fw
C
an cos n
t C
bn sin n t
2
L
L
n=1
n=1
with
&gt;
&gt;
L
1
a0 =
L
an d
bn d
f, cos n
f, sin n
p
t
L
=
p
t
L
=
g t dt
KL
L
1
L
f t cos n
p
t dt, n 2 ;
L
f t sin n
p
t dt, n 2 ;
L
KL
L
1
L
KL
or by rescaling the 2 pKperiodic unit square wave function satisfying
square t =
K1 Kp ! t ! 0
0!t!p
1
which has Fourier series
square t =
4
p
&gt;
n odd
1
sin n t
n
(10 points)
```