MAT 235 Final, Spring 2003
Notre Dame University–Louaize · Ordinary Differential Equations · 10 pages
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NDU
Notre Dame University
MAT 235
Ordinary Differential Equations
Final Exam
Duration: 2 hours
Name:
Section:
Instructor:
Grade:
1) (10 points) Find the family of orthogonal trajectories of the family of curves
x
c
y
ln
, for
x .0
2
2) (20 points) Solve the following differential equations.
a)
0
)
(
)
(
y dy
x
y dx
x
3
b)
x2 y3
x
y
y
3) (20 points) Solve the following differential equations.
4
a)
0
,
2
4
2
x
x
xy
x y
b)
ex
y
y
2
5
4) (8 points) Consider the initial-value problem
0
1, (0)
,0 (0)
2
y
y
x y
y
6
Find its power series solution near the ordinary point
x0 .0
5) (18 points) Consider the differential equation
0
)1
3(
)
(
2
2
y
x
y
x
x
x y
7
a) Show that
x0 0
is a regular singular point.
b) Find the indicial equation.
c) Find two linearly independent solutions.
8
d) Deduce the general solution of the differential equation.
6) (14 points) Solve the following initial-value problem using Laplace transform.
9
0
(0)
(0)
,
t
1
0
1
t
0
y
y
e
y
y
t
7) (10 points) Solve the following linear system.
10
2
1
2
2
1
1
2
2
y
y
y
y
y
y