MATH 267
Due: Jan 25, 2012
ASSIGNMENT # 3
Do the following FIVE problems. Hand in written solutions for grading at the BEGINNING of the lecture on
the due date. Illegible, disorganized or partial solutions will receive no credit. Staple your HW.
1. Find the solution to the following heat equation with boundary conditions and initial conditions:


for all 0 < x < L and t > 0;
 ut (x, t) = uxx (x, t),
u(0, t) = 0, u(L, t) = 0,
for all t > 0;


u(x, 0) = 1
for all 0 < x < L.
(Hint: You may want to look at the course notes ”Solution of the Heat Equation by Separation of
Variables” The Third Step Imposition of the Initial Conditions page 3.)
2. Find the solution to the following wave equation with boundary conditions and initial conditions:


for all 0 < x < L and t > 0;
 utt (x, t) = uxx (x, t),
u(0, t) = 0, u(L, t) = 0,
for all t > 0;


u(x, 0) = 1, ut (x, 0) = f (x)
for all 0 < x < L,
where
(
f (x) =
for 0 < x < L2 ;
for L2 ≤ x < L.
1
−1
(Hint: You may want to look at the course notes ”Solution of the Wave Equation by Separation of
Variables” The Third Step Imposition of the Initial Conditions page 3–4.)
3. (a) Recall eiA = cos A + i sin A and ei(A+B) = eiA eiB . Use these to show the following:
cos(A + B) = cos A cos B − sin A sin B,
sin(A + B) = sin A cos B + cos A sin B.
(b) Use the above result to show the following:
1
cos(A + B) + cos(A − B) ,
2
1
sin A cos B =
sin(A + B) + sin(A − B)
2
cos A cos B =
sin A sin B =
1
cos(A − B) − cos(A + B)
2
4. Fix the constant α2 > 0. Consider the heat equation
ut (x, t) = α2 uxx (x, t),
for all 0 < x < L and t > 0.
Subject to the boundary conditions u(0, t) = 0 and u(L, t) = 0 for all t > 0, solve the initial value problem
if the temperature is initially u(x, 0) = 2 cos(3πx/L). (Hint: You may want to use the result of the above
problem (b).)
5. Consider the boundary value problem for 0 < t < 1,
y 00 (t) + 2y(t) = f (t)
with y(0) = 0 and y(1) = 0, for some nonzero function f (t).
(a) Does this problem has a unique solution? Justify your answer. In other words, suppose there are
solutions y1 (t) and y2 (t) for the problem. Is y1 (t) = y2 (t) for 0 < t < 1? Justify your answer. (Hint.
Consider y = y1 − y2 and plug-in this to the lefthand side of the differential equation. Then, see
this y is not a solution to the original problem, but a solution to a related problem. Is y necessarily
zero?)
1
(b) Suppose f (t) is given by the series
f (t) =
∞
n+1
2 X (−1)
sin (nπt)
π n=1
n
Find a solution to the boundary value problem as a Fourier sine series. (Hint, write an appropriate
Fourier sine series expansion of y(t), and then determine the Fourier coefficients using the given
differential equation. When determining the coefficient, start with n = 1.)
2