MATH 54 − HINTS TO HOMEWORK 13
PEYAM TABRIZIAN
Here are a couple of hints to Homework 13. Make sure to attempt the
problems before you check out those hints. Enjoy!
S ECTION 9.8: T HE MATRIX EXPONENTIAL FUNCTIONS
Again, check out the ‘Systems of differential equations’-handout!
9.8.11. The formula is eAt = X(t)X−1 (0). Remember that you only have
to find the inverse at 0, not in general! Or use eAt = P eDt P −1 , where D is
your matrix of eigenvalues, and P is your matrix of eigenvectors.
9.8.21. The general solution is x(t) = eAt c. Use the initial condition to
solve for c.
S ECTION 10.2: M ETHOD OF SEPARATION OF VARIABLES
110.2.3, 10.2.5. Just solve your equation the way you would usually do (for
5, use undetermined coefficients) and plug in the initial conditions. You
may or may not find a contradiction! If you find 0 = 0, that usually means
there are infinitely many solutions, depending on your constant A or B.
10.2.11. You have to split up your analysis into three cases:
Case 1: λ > 0. Then let λ = ω 2 , where ω > 0. This helps you get rid of
square roots.
Case 2: λ = 0.
Case 3: λ < 0. Then λ = −ω 2 , where ω < 0.
In each case, solve the equation and plug in your initial condition. You
may or may not get a contradiction. Also, remember that y has to be
Date: Tuesday, April 28th, 2015.
1
2
PEYAM TABRIZIAN
nonzero!
10.2.17, 10.2.19. Follow the outline given in the sections ‘Heat equation’
and ‘Wave equation’ in my Partial Differential Equations-Handouts. You
don’t need to worry about Fourier series, as you can just compare the coefficients.
10.2.23. At some point, you should get:
∞
X
An sin(nπx) =
n=1
By ‘comparing,’ you get An =
∞
X
1
sin(nπx)
2
n
n=1
1
.
n2
10.2.28. Just put all the X on the left-hand-side, and all the T on the right00
hand-side. Then you should get XX is constant, equal to λ, which gives you
X 00 = λX, and use this to solve for T .
10.2.33. All they ask you to solve is the differential equation y 00 = 0, with
y(0) = y(L) = 50 for (a), and y(0) = 10 and y(L) = 40 for (b).
S ECTION 10.3: F OURIER SERIES
10.3.7. Just calculate (f g)(−x) = f (−x)g(−x)
10.3.9, 10.3.16. Use the following formulas:
∞
nπx nπx o
a0 X n
f (x)^ +
an cos
+ bn sin
2
T
T
n=1
Z
nπx 1 T
an =
f (x) cos
dx
T −T
T
Z
nπx 1 T
bn =
f (x) sin
dx
T −T
T
Where T is such that f is defined on (−T, T )
10.3.17, 10.3.24. The Fourier series converges to f (x) if f is continuous
+
(x− )
at x, and converges to f (x )+f
if f is discontinuous at x. As for the
2
endpoints T and −T , the fourier series converges to the average of f at
those endpoints.
MATH 54 − HINTS TO HOMEWORK 13
3
10.3.28. For (b), plug in x = 0 in your Fourier series. This is legit because
f is continuous at 0, hence the Fourier series converges to f at 0. For (c),
plug in x = π, here the Fourier series converges to 12 ((−π + )2 + (π − )2 ) =
π2.
S ECTION 10.4: F OURIER COSINE AND SINE SERIES
IMPORTANT NOTE: The book uses the following trick A LOT:
Namely, suppose that when you calculate your coefficients Am or Bm ,
m+1
you get something like: Am = (−1)πm +1 .
Then notice the following: If m is even, then (−1)m+1 + 1 = 0, so
−2
Am = 0, and if m is odd, (−1)m+1 + 1 = −2, and Am = πm
.
So at some point, you would like to say:
f (x)“ = ”
∞
X
Am cos(mx)
m=1,modd
The way you do this is as follows: Since m is odd m = 2k − 1, for
k = 1, 2, 3 · · · , and so the sum becomes:
f (x)“ = ”
∞
X
k=1
−2
cos((2k − 1)x)
π(2k − 1)
10.4.4. π-periodic extension just means ‘repeat the graph of f ’.
The even-2π periodic extension is just the function:
fe (x) =
f (−x)
f (x)
if − π < x < 0
if0 < x < π
The odd-2π periodic extension is just the function:

if − π < x < 0

 −f (−x)
0
ifx = 0
fo (x) =


f (x)
if0 < x < π
And repeat all those graphs!
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PEYAM TABRIZIAN
10.4.7. Use the formulas:
f (x)“ = ”
∞
X
Am cos
πmx T
m=0
where:
Z
1 T
A0 =
f (x)dx
T 0
Z
πmx 2 T
f (x) cos
dx
Am =
T 0
T
10.4.12. Use the formulas:
f (x)“ = ”
∞
X
Bm sin
πmx T
m=0
where:
B0 = 0
2
Bm =
T
Z
T
f (x) sin
0
πmx T
dx
10.4.17. The best advice I can give you is: Read the PDE handout, specifically the section about the heat equation! It outlines all the important steps
you’ll need!