MTHSC 208 (Differential Equations)
Dr. Matthew Macauley
HW 22
Due Wednesday April 15th, 2009
(1) Let Xbe a vector space over C(i.e., the contants are complex numbers, instead of just
real numbers). If {v1, v2}is a basis of X, then by definition, every vector vcan be written
uniquely as v=C1v1+C2v2.
(a) Is the set {1
2v1+1
2v2,1
2v1−1
2v2}a basis of X?
(b) Consider the ODE y00 = 4y. We know that the general solution is y(t) = C1e2t+
C2e−2t, i.e., {e2t, e−2t}is a basis for the solution space. Use (b), and the fact that
ex= coshx+ sinhxto find a basis for the solution space involving hyperbolic sines
and cosines, and write the general solution using these functions.
(2) We will find the function u(x, t), defined for 0 ≤x≤πand t≥0, which satisfies the
following conditions:
∂u
∂t = 9 ∂2u
∂x2, u(0, t) = u(π, t) = 0, u(x, 0) = sin x+ 3 sin 2x−5 sin 7x.
(a) Assume that u(x, t) = f(x)g(t). Plug this back into the PDE and separate variables
to get the eigenvalue problem (set equal to a constant λ). Solve for g(t), f(x), and λ.
(b) Using your solution to (a), find the general solution to the PDE
∂u
∂t = 9 ∂2u
∂x2
subject to the Dirichlet boundary conditions:
u(0, t) = u(π, t)=0.
(c) Solve the initial value problem, i.e., find the particular solution u(x, t) that satisfies
u(x, 0) = sin x+ 3 sin 2x−5 sin 7x.
(d) What is the steady-state solution, i.e., lim
t→∞ u(x, t)?
(3) Find the function u(x, t), defined for 0 ≤x≤πand t≥0, which satisfies the following
conditions:
∂u
∂t = 9 ∂2u
∂x2, u(0, t) = u(π, t) = 0, u(x, 0) = x(π−x).
Note: The general solution will be exactly the same as in the previous problem. All you
need to do again is Part (c) and (d) for this new initial condition, u(x, 0) = x(π−x).
Additionally, sketch the bar and its initial heat distribution.
(4) We will find the function u(x, t), defined for 0 ≤x≤πand t≥0, which satisfies the
following conditions:
∂u
∂t =∂2u
∂x2, ux(0, t) = ux(π, t) = 0, u(x, 0) = 4 + 3 cos x+ 8 cos 2x .
(a) Assume that u(x, t) = f(x)g(t). Plug this back into the PDE and separate variables
to get the eigenvalue problem (set equal to a constant λ). Solve for g(t), f(x), and λ.
(b) Using your solution to (a), find the general solution to the PDE
∂u
∂t =∂2u
∂x2
subject to the Neumann boundary conditions:
ux(0, t) = ux(π, t)=0.
