In all of the problems below recognize the Sturm-Liouville problem as one solved before and take for granted its eigenvalues and eigen functions. If the problem is non-homogeneous transform it first into a homogeneous 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 equation 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).

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