MAT 235 Final, Fall 2004
Notre Dame University–Louaize · Ordinary Differential Equations · 12 pages
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NDU
MAT 235
Ordinary Differential Equations
Final Exam
Duration: 120 Minutes.
Name: ___________________________________
Section: __________________________________
Instructor: ________________________________
Grade: ___________________________________
3/13
1-
(9 points) Solve each of the differential equations:
a)
0
cos
2
sin
2
y dy
x dx
x
y
4/13
b)
2
3
1
2
3
y
x
y
x
dx
dy
5/13
2- (5 points) Given
t
u du
u
t
t
f
0
5
,
sin
find Laplace transform of
f x
3- (7 points) Given
2
2
9
2
2
s
e
s
s
s
F
, find Laplace inverse of s
F
6/13
4-
(10 points) Find the general solution of the differential equation
x e x
y
y
y
2
4
4
4
for
x 0
7/13
5-
(8 points) Find the general solution of the system
y
x
y
y
x
x
2
4
8/13
6-
(7 points) Find the general solution of
0
2
1
2
1
2
y
y
x
y
x
. For
x 1 0
(Hint: you can use the substitution
u x 1
) .
9/13
7-
(12 points) Use Laplace transforms to solve the following initial-value problem
h t
y
y
where
t
0
t
0
t
h t
with
0
0
,2
0
y
y
10/13
8- (6 points) Find a family of oblique trajectories that interest the family of hyperbolas
xy c
at angle 45o
11/13
9- (16 points) Use the Laplace transform to find the solution of the given linear system
that satisfies the given initial conditions
2 0
y
y
x
x
0
4
8
5
7
y
y
x
x
with
x 0 1
and
1
0
y
12/13
(20 points) Given the differential equation
.0
2
4
3
2
2
x y
x y
x
x y
Show that
x0 0
is a regular singular point, then use the method of Frobenius to find
solutions of the given equation in some interval
.
0
x R
13/13