solve the problems attached
HW 5
In all of the problems below recognize the Sturm-Liouville problem
as one solved before and take for granted its eigenvalues and eigenfunc-
tions. If the problem is non-homogeneous transform it first into a homo-
geneous one as follows: make the substitution u(x, t) = U(x, t) + h(x)
or y(x, t) = Y (x, t)+h(x), impose conditions on h(x) so that the equa-
tion in Y or U become homogeneous and then find h(x). Solve then
the homogeneous problem and add h(x) to the final answer.
(
1
) Wave equation with non-zero initial velocities.
ytt = yxx
y(x, 0) = x2, 0 < x < 1
yt(x, 0) = x, 0 < x < 1
y(0, t) = y(1, t) = 0, t > 0
(2) Wave Equation with left end-point kept at positive displacement
ytt = yxx
y(x, 0) = x, 0 < x < π
yt(x, 0) = 0, 0 < x < π
y(0, t) = 2, y(π, t) = 0, t > 0
(3) Heat equation with an external source of heat and prescribed
non-zero temperatures at end-points
ut = uxx + x
u(x, 0) = x, 0 < x < 6
u(0, t) = 1, u(6, t) = 7, t > 0
(4) Heat equation with a constant heat loss and prescribed heat
flows at the end-points
ut = 4uxx − 8
u(x, 0) = x, 0 < x < 1
ux(0, t) = 0, ux(1, t) = 2, t > 0
(5) Heat Equation in a sphere
ut = urr +
2
r
ur
u(r, 0) = r, 0 < r < 1
u(0, t) = 0, u(1, t) + ur(1, t) = 0, t > 0
Hints: proceed like the example in class. Multiply both sides
of the PDE by r and make the substitution U(r, t) = ru(r, t).
1
2 HW 5
You should obtain a heat equation in U(r, t) that looks like one in
cartesian coordinates and with Newmann homogeneous bound-
ary conditions (prescribed values of derivatives)
Solve the following Cauchy-Euler Equations
(6) (a) x2y′′ + 3xy′ + 5y = 0
(b) x2y′′ + 5xy′ + 4y = 0
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