MATH 3321
Final Exam — Sample Questions
1.
y
2
=
Cx
3

3
is the general solution of a differential equation. Find the equation.
Answer
y
=
3
y
2
+ 9
2
xy
2.
Find the general solution of
x y
+ 3
y
=
cos 2
x
x
2
.
Answer
y
=
sin 2
x
2
x
3
+
C
x
3
.
3. Find the general solution of 2
y
=
y
2
+ 3
4
y
+
xy
.
Answer
y
2
=
C
(4 +
x
)

3
4. Find the general solution of
y
+ 6
y
+ 9
y
= 0.
Answer
y
=
C
1
e

3
x
+
C
2
xe

3
x
.
5. Determine a fundamental set of solutions of
y
+ 4
y
+ 13
y
= 0.
Answer
{
e

2
x
cos 3
x, e

2
x
sin 3
x
}
6. The function
y
=

2
e

3
x
sin 2
x
is a solution of a second order, linear, homogeneous differ
ential equation with constant coefficients. What is the equation?
Answer
y
+ 6
y
+ 13
y
= 0
7. The function
y
=

2
e
3
x
+ 4
xe
3
x
is a solution of a second order, linear, homogeneous
differential equation with constant coefficients. What is the equation?
Answer
y

6
y
+ 9
y
= 0
8. Given the initialvalue problem
y

7
y
+ 12
y
= 0;
y
(0) = 3
, y
(0) = 0.
(a) Find the general solution of the differential equation.
(b) Find the solution that satisfies the initial conditions.
Answer
(a)
y
=
C
1
e
3
x
+
C
2
e
4
x
.
(b)
y
= 12
e
3
x

9
e
4
x
.
9. Given the oneparameter family
y
3
=
Cx
2
+ 4.
(a) Find the differential equation for the family.
(b) Find the differential equation for the family of orthogonal trajectories.
1
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(c) Find the family of orthogonal trajectories.
Answer
(a)
y
=
2
y
3

8
3
xy
2
.
(b)
y
=

3
xy
2
2
y
3

8
.
(c) 3
x
2
y
+ 2
y
3
+
Cy
+ 16 = 0.
10. A certain radioactive material is decaying at a rate proportional to the amount present. If a
sample of 100 grams of the material was present initially and after 2 hours the sample lost
20% of its mass, find:
(a) An expression for the mass
A
(
t
)
of the material remaining at any time
t
.
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 Fall '08
 morgan
 Math, Linear Algebra, Eigenvalue, eigenvector and eigenspace, Complex number, Orthogonal matrix, general solution

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