Download Final Exam – Sample Questions 1. y = Cx

Final Exam – Sample Questions
1. y 2 = Cx3 − 3 is the general solution of a
differential equation. Find the equation.
Answer:
2+9
3y
y0 =
2xy
2. Find the general solution of
x y0 + 3 y =
Answer:
cos 2x
x2
sin 2x
C
y=
+ 3.
2x3
x
3. Determine a fundamental set of solutions
of
y 00 − 2 y 0 − 15 y = 0
Answer:
n
e5x, e−3x
o
4. Find the general solution of
y 00 + 6 y 0 + 9 y = 0
Answer:
y = C1e−3x + C2xe−3x.
5. The function y = −2 e−3x sin 2x is a solution of a second order, linear, homogeneous differential equation with constant
coefficients. What is the equation?
Answer:
y 00 + 6y 0 + 13y = 0
y2 + 3
0
6. Find the general solution of 2y =
.
4y + xy
Answer:
y 2 = C(4 + x) − 3
7. The function y = −2 e3x + 4 xe3x is a
solution of a second order, linear, homogeneous differential equation with constant
coefficients. What is the equation?
Answer:
y 00 − 6y 0 + 9y = 0
8. Given the initial-value problem
y 00 − 7 y 0 + 12 y = 0;
y(0) = 3, y 0(0) = 0
(a) Find the general solution of the differential equation.
(b) Find the solution that satisfies the initial
conditions.
Answer:
(a) y = C1e3x + C2e4x.
(b) y = 12e3x − 9e4x.
9. Given the one-parameter family y 3 = Cx2+
4.
(a) Find the differential equation for the family.
(b) Find the differential equation for the family of orthogonal trajectories.
(c) Find the family of orthogonal trajectories.
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 3 hours the
sample lost 30% of its mass, find:
(a) An expression for the mass A(t) of the
material remaining at any time t.
(b) The mass of the material after 8 hours.
(c) The half-life of the material.
7 t/3
Answer: (a) A(t) = 100
(b)
10
7 8/3
−3 ln 2
A(8) = 100
(c) t =
10
ln 7/10
11. Scientists observed that a colony of penguins on a remote Antarctic island obeys
the population growth law. There were
1000 penguins in the initial population and
there were 3000 penguins 4 years later.
(a) Give an expression for the number P (t)
of penguins at any time t.
(b) How many penguins will there be after
6 years?
(c) How long will it take for the number of
penguins to quadruple?
t ln(3)
4
1000e
Answer: (a) P (t) =
(b) P (6) = 1000 (3)3/2.
4 ln(4)
(c) t =
years.
ln(3)
= 1000 (3)t/4
12. A disease is infecting a herd of 1000 cows.
Let P (t) be the number of sick cows
t days after the outbreak. Suppose that
50 cows had the disease initially, and suppose that the disease is spreading at a rate
proportional to the product of the time
elapsed and the number of cows who do
not have the disease.
(a) Give the mathematical model (initialvalue problem) for P .
(b) Find the general solution of the differential equation in (a).
(c) Find the particular solution that satisfies
the initial condition.
dP
Answer: (a)
= kt(1000 − P ), P (0) =
dt
50
(b) P (t) = 1000 − Ce−kt2/2
(c) P (t) = 1000 − 950e−kt2/2
13. The functions y1 (x) = −2x3+2 ln x, y2(x) =
3x−2 + 2 ln x, y3 (x) = 2x3 − 5x−2 + 2 ln x
are solutions of a second order nonhomogeneous linear differential equation y 00 +
p(x)y 0 + q(x)y = f (x).
(a) Find a fundamental set of solutions of
the reduced equation; justify your answer using the Wronskian.
(b) Give the general solution of the reduced
equation.
(c) What is the reduced equation?
(d) Give the general solution solution of the
nonhomogeneous equation.
14. Find a particular solution of the nonhomogeneous differential equation
y 00 − 6 y 0 + 8 y = 4 e4x.
Answer:
z = 2xe4x.
15. Give the form of a particular solution of the
nonhomogeneous differential equation
y 00 − 8 y 0 + 16 y = 2 e4x + 3 cos 4x − 2.
Answer: z = Ax2e4x+B cos 4x+C sin 4x+
D.
16. Given the differential equation
e2x
00
0
y − 4y + 4y =
x
(a) Give the general solution of the reduced equation.
(b) Find a particular solution of the nonhomogeneous equation.
Answer:
(a) y = C1e2x + C2xe2x
(b) z = −xe2x + x e2x ln x or z = x e2x ln x
17. Find the general solution of
y 00 −
Answer:
4 0
6
4
y + 2 y = 2.
x
x
x
y = C1x2 + C2x3 +
2
3
18. Find the general solution of
y 00 + 4 y = 2 tan 2x
Answer:
y = C1 cos 2x + C2 sin 2x−
1 cos 2x ln
[sec 2x + tan 2x]
2
3
4s + 3
19. If F (s) = 2 + 2
, find L−1[F (s)].
s
s +4
Answer:
f (x) = 3x + 4 cos 2x + 3
2 sin 2x
20. Find the Laplace transform of the solution
of the initial-value problem
y 00 − y 0 − 6 y = 3; y(0) = 5, y 0(0) = 0
Answer:
3
5s − 5
Y =
+ 2
.
2
s(s − s − 6)
s −s−6
21. Find L[f (x)] if
f (x) =
(
x2 + 2x
x
0≤x<4
x≥4
Answer:
2
2
−4s 2
−4s 1
−4s 1
F (s) = 3 + 2 −e
−9e
−20e
s
s
s3
s2
s
5
1
1
s+1
−3s
−3s
−3s
22. Let F (s) = 2 −2 e
+3 e
+2e
.
s
s2
s
s2 + π 2
Find L−1 [F (s)] = f (x).
Answer:
f (x) =


 3x + 9 − 2 cos πx −
5x,
1
sin πx,
π
23. Given the initial-value problem
y 0 − 4 y = 2 e−2x, y(0) = 3.
(a) Find the Laplace transform of the solution.
(b) Find the solution by finding the inverse
Laplace transform of your answer to (a).
2
3
+
.
(s + 2)(s − 4)
s−4
4x − 1 e−2x.
(b) y = 10
e
3
3
Answer:
(a) Y =
24. Given the system of equations
x +2y −z = 1
2x +5y −4z = 3
−2x −2y −2z = 0
(a) Write the augmented matrix for the
system.
(b) Reduce the augmented matrix to rowechelon form.
(c) Give the solution set of the system.


1
2 −1 1


Answer: (a)  2
5 −4 3 .
−2 −2 −2 0


1 2 −1 1


 0 1 −2 1 .
0 0 0 0
(c) x1 = −1 − 3a,
a, a arbitrary.
x2 = 1 + 2a,
(b)
x3 =
25. Given the system of equations
x −2y
=1
x −y +kz = 3
y −2z = k
The value(s) of k, if any, such that the
system has infinitely many solutions is (are):
Answer:
No values of k.
26. Find the values of  λ, (if any) such that
λ
0
3


A= 0
1
λ  is nonsingular.
−2 −1 −5
Answer:
λ 6= 2, 3




1
2
0

27. The matrix
2
5 −3  is non−2 −4
1
singular. Calculate A−1.

A=
Answer:

A−1 = 
−7 −2 −6

4
1
3 .
2
0
1
28. Use Cramer’s rule to find the value of x2:
2x1 − x2 +
3x3 = 4
x2 + 2x3 = −2
x1
+
x3 = 1
Answer:
x2 = −2
29. Determine whether the vectors
v1 = ( 1, −3, 2 ), v2 = ( 0, −2, −2 ),
v3 = ( 1, −5, 0 ), v4 = ( 0, 4, 4 )
are linearly dependent or linearly independent. If they are linearly dependent, find
the maximal number of independent vectors.
Answer: Linearly dependent; the maximum number of independent vectors is 2.
30. Find
the eigenvalues
and eigenvectors of


2
2 −6


 2 −1 −3 . Hint: 6 is an eigenvalue.
−2 −1
1
Answer:




2
1




λ1 = 6,  1  ; λ2 = −2,  −2  ;
−1
0


0


λ3 = −2,  3 
1
31. Find
the eigenvalues
and eigenvectors of


−2
1 −1


4 . Hint: 1 is an eigenvalue.
 3 −3
3 −1
2
Answer:

1



λ1 = 1,  1  ;



−2,  −1 
−1
0

1
λ2 = λ3 =
32. Find
the eigenvalues
and eigenvectors of


2 −1 −1


1 −1 . Hint: 1+i is an eigenvalue.
 2
0 −1
1
Answer:

1




λ1 = 2,  −1  ;

0





i,  0  + i  −1  ;
1

0


i  −1 
1

0

1
1
λ2 = 1 +

1



λ3 = 1 − i,  0  −
1
33. Find the general solution of
y (4) − 6 y 000 + 17 y 00 − 28 y 0 + 20 y = 0
Answer: y = C1e2x+C2xe2x+C3ex cos 2x+
C4ex sin 2x.
34. A particular solution of
y (4) + 5 y 00 − 36 y = 2x + 3e2x − 2 sin 3x
will have the form
Answer: z = Ax+B+Cxe2x +Dx cos 3x+
Ex sin 3x.
35. Find the solution
of the

 initial-valueprob
1 1 2
2




lem x0 =  0 2 2 x; x(0) =  0 .
−1 1 3
1
HINT: 2 is a root of the characteristic
polynomial.
Answer:




1
0



x(t) = 2e2t 
 1  − et  2 .
0
−1
36. Find a fundamental set of solutions of
x0 =
3 −5
2
1
!
x
Answer:
"
x1 = e2t cos 3t
"
x2 = e2t cos 3t
1
2
!
3
0
!
− sin 3t
+ sin 3t
3
0
1
2
!#
,
!#


1
1
1

1 −1 x.
0 −1
1
is a root of the characteristic

37. Find the general solution of x0 =  2
HINT: 2
polynomial.
Answer:


C3 e2t 






0
−3



x(t) = C1e−t 
 4 +C2e2t  1 +
−1
2



1
0



0  + te2t  1 .
1
−1
38. Find a fundamental set of solutions of


5 4 2

x0 = 
 4 5 2 x
2 2 2
HINT: 10 is a root of the characteristic
polynomial.




0
1
2 







t
t
10t
Answer:
e  1 , e  0 , e
 2  .



−2
−2
1 




