MAT 235 Final, Fall 2003
Notre Dame University–Louaize · Ordinary Differential Equations · 13 pages
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NDU
MAT 235
Ordinary Differential Equations
Final Exam
Wednesday February 4, 2004
Duration: 2 hours
Name: ________________
Section: _______________
Instructor: ____________
Grade: ________________
1/12
Please note that you have 10 questions and 12 pages
1) (10 points) Solve the following:
a)
0
2
2
2
dy
y
x
xydx
with
y 0 1
b)
2
3
2
3
2
2
y
x
y
x
dx
dy
2/12
2) (10 points) Find the orthogonal trajectories of the family of circles
2
2
2
c
y
c
x
.
3/12
3) (4 points) Find the Laplace inverse of the following function
3
ln
s
s
F s
4/12
4) (8 points) Solve
1
1
y
x y
for
x 1
5/12
5) (10 points) Solve
x
y
y
sec3
9
6/12
6) (8 points) Use Laplace transform to solve
1
0
1
0
2
7
6
t
t
y
y
y
with
0
0
0
y
y
7/12
7) (10 points) Solve the following initial-value problem by using Laplace transform.
0
2
3
0
2
2
2
1
2
1
1
y
y
y
y
y
y
with
1
1 0
y
and
2
2 0
y
8/12
8) (8 points) Use the eigenvalue-eigenvector method to solve
2
1
2
2
1
1
3
7
2
2
y
y
y
y
y
y
9/12
9) (12 points) Given that
x0 0
is an ordinary point of the differential equation
0
2
2
1
2
y
xy
y
x
. Find the general power series solution in powers of x.
10/12
11/12
10)
(20 points) Given
0
1
4
8
4
2
2
2
y
x
x y
x y
for
x 0
.
a) Show that
x0 0
is a regular singular point.
b) Find the indicial roots.
c) Find the generalized power series solution in powers of x.
12/12