Homework 1
1
May 16, 2011
1.1
1 a. 1
b. 2
c. 1
d. 4
e. 5
2e We calculate
ux = ex cos(y) + a
uxx = ex cos(y)
uy = ex sin(y) + b
uyy = ex cos(y)
so
uxx + uyy = ex cos(y) ex cos(y) = 0
3b For u = 7 sin(x ct),
ut = 7c cos(x ct),
ux = 7 cos(x ct)
so
ut + cux =
Homework 2
May 16, 2011
1
Recall, the denition of linearity for an operator L[u]:
Denition 0.1 (Linearity of PDE operator). A PDE in operator form, i.e. expressed as L[u] = f
like in class, is linear if for any two solutions u1 and u2 and any real number
Homework 3
1
May 16, 2011
See the end for a proof of something used in class (exercise 36)!
1.6
3,24 We wish to solve the PDE
y 2 ux + x2 uy = 0,
u = u(x, y)
so we assume that there is a product solution, u(x, y) = X(x)Y (y). Substituting gives us
y 2 X Y
Homework 4
1
May 16, 2011
1.7
3 Solve
y + y = 0
(0.1)
y (0) = 0
(0.2)
y() = 0
(0.3)
If we guess a solution to equation (0.1) of the form y = erx , we see that y = rerx , and
y = r2 erx , so when we substitute, we get the characteristic equation
y + y = r2
Homework 5
1
May 16, 2011
Note again that having the answer in any and all of these problems is not terribly impressive
from a grading standpoint, which is why we provide solutions: it is knowing and doing the work in
between the problem and answer which