MAT 235 Final, Spring 2004
Notre Dame University–Louaize · Ordinary Differential Equations · 12 pages
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NDU
MAT 235
Ordinary Differential Equations
Final Exam
Duration: 2 hours
Name: ________________
Section: _______________
Instructor: ____________
Grade: ________________
MAT 235 – Final Exam; Friday June 18, 2004
Name:
Instructor:
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Please note that you have 10 questions and 11 pages
(your mobile must be turned off and unseen)
1) (8 points) Solve the differential equation
x3y4
y
xy
, for
x 0
.
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2) (6 points) Solve the initial-value problem
1
xy
y
dx
x dy
;
with
y 1 3
.
Hint: Let
v
xy
1
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3) (8 points) Solve
0
cos
2
4
y dy
xy
x dx
y
x
.
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4) (6 points) Solve the differential equation
0
2
2
3
y
xy
x y
x y
, for
x 0
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5) (10 points) Solve the differential equation
2
1
2
x
e
y
y
y
x
; for
x 1
.
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6) (8 points) Find the function t
x
such that
t
0
u x u du
t
t
x t
sin
2
.
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7) (12 points) Use Laplace transform to solve
2
1
2
2
1
1
2
5
2
4
y
y
y
y
y
y
;
2
1 0
y
;
2
2 0
y
.
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8) (12 points) Use Laplace transform to solve
4
;0
4
0
25;
25
2
2
t
t
y
dt
d y
;
0
0
0
y
y
.
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9) (10 points) Use the eigenvalue-eigenvector method to solve
2
1
1
6y
y
y
2
1
2
2y
y
y
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10)
(20 points) Given
0
4
1
2
2
y
x
xy
x y
, for
x 0
.
a) Show that
x0 0
is a regular singular point.
b) Find the indicial roots.
c) Use the method of Frobenius to find the generalized power series solution in powers
of x.
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