MATH 285: Worksheet 11
April 9, 2015
9.5/9.6 Main ideas
The Method of Separation of Variables for Partial Differential Equations
Two Important PDEs
1. Section 9.5: Heated Rods and the One-Dimensional Heat Equation
Guiding principle:
Differential Equation:
Conditions:
2. Section 9.6: Vibrating Strings and the One-Dimensional Wave Equation
Guiding principle:
Differential Equation:
Conditions:
Section 9.7: The Two-Dimensional Heat and Wave Equations
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The Method of Separation of Variables
1.
2.
3.
4.
5.
6.
Recall
1. x00 + λx = 0, x(0) = x(L) = 0. Solution:
2. x00 + λx = 0, x0 (0) = x0 (L) = 0. Solution:
Some Helpful Integrals
Z 2
cos2 (2πx) dx = 1
•
0
Z
•
2
2
cos (2πx) cos
0
2
•
L
Z
L
sin
0
nπx L
nπx 2
dx =
dx =
0
4
πn
0 n 6= 8
1
n=8
2
n even
n odd
Z
nπx 2 L
0
n even
•
x cos
dx =
−4L
n odd
L 0
L
π 2 n2
Z π
0
n even
sin2 x sin (nx) dx =
•
4
n odd
n(4−n2 )
0
2
Example:
5ut = uxx
0 < x < 10
t>0
ux (0, t) = ux (10, t) = 0
3
u(x, 0) = 4x
Practice Problems
1. A
is modelled by the differential equation below. Solve and explain
what the conditions and any constants mean.
10ut = uxx
0<x<5
t>0
u(0, t) = u(5, t) = 0
u(x, 0) = 25
2. A
is modelled by the differential equation below. Solve and explain
what the conditions and any constants mean.
ytt = 4yxx
0<x<π
t>0
y(0, t) = y(π, t) = 0
y(x, 0) = sin x
yt (x, 0) = 1
is modelled by the differential equation below. Solve and explain
3. A
what the conditions and any constants mean.
ytt = 25yxx
0<x<π
t>0
y(0, t) = y(π, t) = 0
y(x, 0) = yt (x, 0) = sin2 x
4. A
is modelled by the differential equation below. Solve and explain
what the conditions and any constants mean.
3ut = uxx
0<x<2
t>0
ux (0, t) = ux (2, t) = 0
u(x, 0) = cos2 (2πx)
5. Find the Fourier sine and cosine series for f (t) = t(π − t), 0 < t < π and sketch the
two graphs to which these series converge.
6. Find a formal Fourier series to x00 + 2x = t2 , x(0) = x(π) = 0.
7. Determine if resonance occurs in the system modelled below and find xsp and check
for resonance.
x00 + 4π 2 x = F (t)
where F (t) is the odd function of period 2 with F (t) = 2t for 0 < t < 1
An Ending Thought: Perserverance is the hard work you do after you get tired
of doing the hard work you already did.
– Newt Gingrich
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