MAT 235 Final, Spring 2008
Notre Dame University–Louaize · Ordinary Differential Equations · 9 pages
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NDU
MAT 235
Ordinary Differential Equations
Final Exam
Duration: 2 hours
Name: ________________
Section: _______________
Instructor: ____________
Grade: ________________
(1)
1) (12 points) Solve
a)
0
tan
dy
x dx
y
x
y
b)
3
2
2
5
x y
x
y
dx
dy
(2)
2) (10 points) Solve
2
1
2
x
e
y
y
y
x
.
(3)
3) (15 points) Find the Laplace transform
F s
of each of the following functions t
f
.
a)
t
ydy
y
t
f
0
sin
b)
t
ydy
y
t
t
f
0
5 cos
.
c)
t
e
t
f
t
2 1
(Recall that, if the Laplace Transform of
F s
f t
, then the Laplace transform of
s
f y dy
t
f t
).
(4)
4) (15 points) Find the inverse Laplace Transform t
f
of each of the following functions
F s
.
a)
1
2
1
2
2
s
s
s
F s
b)
s
F s
arctan 1
c)
2
2
1
ln
s
s
F s
(5)
5) (10 points) Use the Laplace transform to solve
e t
y
y
y
4
2
with
y 0 2
, and
1
0
y
.
(6)
6) (10 points) Solve
2
0
,0 and
0
with
0
4
2
y
x
y
y
x
e
y
x
x
(7)
7) (18 points) Use the method of Frobenius to find the general solution near
x 0
of
0
2
xy
y
xy
.
(8)
8) (10 points) Find the generalized power series solution near
x 0
of
0
2
2
1
2
y
xy
y
x
(Hint: the point
x 0
is an ordinary point of the given equation).