Hint for Problem 9.24
May 9, 2011
The problem is
Txx + Tyy = sin(x)
T (x = 0) = T (x = l) = T (y = 0) = 0, T (y ) bounded.
The PDE can be reduced to a homogenous one by the substitution
U = T + A sin(x)
Uxx = Txx A 2 sin(x)
Uyy = Tyy .
Substitute into the

Solution to Problem 4.15
March 14, 2011
Here, R(x) = 1, P (x) = x 1, Q(x) = x.
We therefore have
f (s + k ) = (s + k )(s + k 1) (s + k ) = (s + k )(s + k 2)
(1)
gn (s + k ) = (s + k 1) if n = 1 and 0 if n > 1.
(2)
and
The indicial equation is
f (s) = (s)(

Solutions to Homework 11
May 23, 2011
9.52
The problem is
1
wtt , for 0 < x < L, 0 < y < L
c2
w(x = 0) = w(x = L) = w(y = 0) = w(y = L) = 0; w bounded.
wxx + wyy =
Looking for separable solutions by substituting
w(x, y, t) = X (x)Y (y )T (t),
we get
1
XY

Solutions to Final Exam
May 24, 2011
problem 1
a
At steady state, we have
Txx + C = 0 , for 0 < x < L
T (x = 0) = 0, Tx (x = L) = 0
The general solution to the ODE is
T=
C 2
x + ax + b
2
The boundary conditions give us b = 0, a = CL, so we nally have
x
)

Engineering Analysis Final Exam
1] Consider a rod with a uniform internal heat source that is kept at the background temperature at one
end and insulated on the other, i.e.
2
T
1 T for
C = 2
t 0, 0 x L
2
x
t
where 2 and C are positive constants, with bou

Exam 2 Solutions
1]
(a) If a solution is in the form
2
1 x n n1 A n x
n 2
n= 2
n= 0
2x n A n x
n 1
n= 0
n=0
n
, then substituting into the equation gives us
n
2 A n x =0 , or
n= 0
n= 2
y x = A n x
n n1 A n x n2 2 2n n n 1 A n x n=0
, or
n 2 n 1 An2 x n

Engineering Analysis Exam 2
1] (a) By assuming solutions in the form of power series about x = 0, find two independent solutions
for the differential equation
2
dy
2dy
1 x 2 2x 2y =0 .
dx
dx
(b) Determine the convergence radius of the two power-series so

Exam 1 solutions
(a)
ht
d 2 x c dx k
x=
2
m dt m
m
dt
(b)
This is a linear second-order ordinary differential equation with constant coefficients.
(c)
We assume a solution of the form x = Cert, giving for r the quadratic equation:
r2 + (c/m)r + (k/m) =

Engineering Analysis Exam 1
Balance of forces for a damped linear oscillator gives
m
d2 x
dx
=c kx h t
2
dt
dt
where c, k, m are positive numbers.
(a) Write this differential equation in standard form.
(b) Classify this differential equation. Be as speci