Math. 401, Sec. 500 Spring, 2005
Homework 7, due March 9
1. Classify these equations as linear homogeneous, linear nonhomogeneous, or nonlinear.
(Here ut≡∂u/∂t,etc.)
(a) utt −uxx =cos(x−t)
(b) ut+u2ux=0
(c) ut+3t
2
u=0
(d) utt −uxx =−m2u
2. Why do we not make a distinction between “homogeneous” and “nonhomogeneous”
for nonlinear equations? Hint: Try to classify the equation y00 +(cosy)
2=1.
3. [Schaum’s, p. 46, Ex. 2.52] Find the steady-state temperature in a bar whose ends
are located at x=0andx= 10, if these ends are kept at 150◦C and 100◦C respectively.
4. Suppose that the boundary conditions (“BC:”) on p. 64 of the notes (and corresponding
lecture on the heat equation with fixed end temperatures) are replaced by
∂u
∂x(t, 0) = F1,∂u
∂x(t, 1) = F2.
(That is, the heat flux through each end of the bars is held constant.) What happens
when you attempt to find a steady-state solution as on p. 66? Distinguish between the
two cases F1=F2and F16=F2. Can you give a physical explanation for your results?
