Math 220, Multiple Choice Sample
We discuss the distribution of questions on Review day.
Linear first order:
1. Which one of the following differential equations is linear.
(a)
ty 0 − (t + 1)y = tan t
(b)
ty 0 − (t + 1)y = y 3
(c)
3t2 y 2 + sin t + 2t3 yy 0 = 0
(d)
ty 0 − (t + 1)y = tan y
2. Which of the following is an integrating factor for the differential equation?
ty 0 − (2t + 3)y = 2 sec t,
0 < t < π/2
Do not solve the differential equation. Use the standard form of the linear equation.
(b)
2t + 3
t
−2t 3
µ(t) = e t
(c)
µ(t) = e−2t t−3
(d)
µ(t) = e−2t
1
µ(t) = 3
t
(a)
(e)
µ(t) = −
3. Which of the following is an integrating factor for the differential equation?
t2 y 0 + 3ty = 2 sin t, t > 0
Do not solve the differential equation. Use the standard form of the linear equation.
(a)
µ(t) = e3t
(b)
µ(t) = t−3
(c)
µ(t) = t3
(d)
µ(t) = tet
1
4. Solve the linear equation ty 0 + 3y = t−3 , t > 0, using the integrating factor µ(t) = t3 for
standard form of the linear equation.
(a)
t−3
(b)
t−3 ln t + ct−3
(c)
t−3 + c ln t
(d)
t−3 ln t
5. Solve the linear equation y 0 + 3y = e−t , using the integrating factor µ(t) = e3t .
(a)
e−3t
(c)
e−t
+ ce−3t
2
e−3t + ce−t
(d)
e−t + e−3t
(b)
Exact equations:
1. Which one of the following equations is exact?
(a)
2x3 y 3 dx + 3x4 y 2 dy = 0
(b)
x3 y 3 dx + 3x4 y 2 dy = 0
(c)
4x3 y 3 dx + 3x4 y 2 dy = 0
(d)
4x3 y 3 dx + x4 y 2 dy = 0
2. Which one of the following equations is exact?
(a)
cos(x + y) − sin x + cos(x + y)y 0 = 0
(b)
cos(x + y) − sin y + cos(x + y)y 0 = 0
(c)
cos y + cos x + cos(x + y)y 0 = 0
(d)
cos(x + y) − sin x − cos(x + y)y 0 = 0
2
Recognition of first order odes.
1. Consider the equation 2xy 3 + 3x2 y 2 y 0 = 0. Check ALL that apply.
(a)
The equation is linear.
(b)
The equation is Exact.
(c)
The equation is separable.
(d)
The equation is Bernoulli
2. Consider the equation ey cos x dx + sin y dy = 0. Check ALL that apply.
(a)
The equation is linear.
(b)
The equation is Exact.
(c)
The equation is separable.
(d)
The equation is Bernoulli
3. Consider the equation ty 0 − 3y = et . Check ALL that apply.
(a)
The equation is linear.
(b)
The equation is separable.
(c)
The equation is Bernoulli
Explicit solution:( Separable)
1. Solve the initial value problem y 0 = √
(a)
(b)
(c)
(d)
xy 3
,
x2 + 1
12
1
√
−1 + 2 x2 + 1
12
1
√
y=
3 − 2 x2 + 1
12
1
√
y=−
3 − 2 x2 + 1
12
1
√
y=
−1 + 2 x2 + 1
y=−
3
y(0) = −1
2. Solve the initial value problem y 0 (y + 1) = ex + x2 ,
y(0) = 2
r
(a)
(b)
(c)
(d)
2x3
y = −1 − 7 + 2ex +
3
r
2x3
y = 1 + −1 + 2ex +
3
r
2x3
y = 1 − −1 + 2ex +
3
r
2x3
y = −1 + 7 + 2ex +
3
Tank problems:
1. A 800-gallon tank initially contains 30 pounds of salt. Water enters the tank at the rate of 3
gal/min with concentration 4 lb/gal of salt in it. The well-mixed solution leaves the tank at
the same rate of 3 gal/min. Which of the initial value problems below models the change of
the amount of salt Q(t) inside the tank at any time t?
(a)
(b)
(c)
(d)
3Q
800
3Q
Q0 (t) = 4 −
800
4Q
Q0 (t) = 3 −
800
3Q
Q0 (t) = 12 −
800
Q0 (t) = 12 −
Q(0) = 800
Q(0) = 30
Q(0) = 30
Q(0) = 30
2. A 800-gallon tank initially contains 30 pounds of salt. Water enters the tank at the rate of 3
gal/min with concentration 4 lb/gal of salt in it. The well-mixed solution leaves the tank at
the same rate of 3 gal/min. What is the limiting value of Q?
(a)
800 lb
(b)
4 lb
(c)
12 lb
(d)
3200 lb
4
Modified Guess for particular solution
1. It is easy to see that the homogeneous solution to y 00 + 2y 0 + 5y = 0 is
yh = C1 e−t cos(2t) + C2 e−t sin(2t). Now find a modified guess yp for y 00 + 2y 0 + 5y = e−t
(a)
yp = Ae−t
(b)
yp = Ate−t
(c)
yp = Ae−t + Bt
(d)
yp = Ae−t cos(2t) + Be−t sin(2t)
2. It is easy to see that the homogeneous solution to y 00 + 4y = 0 is
yh = C1 cos(2t) + C2 sin(2t). Find a modified guess yp for y 00 + 4y = e2t + cos(2t)
(a)
yp = Ae2t + B sin(2t) + C cos(2t)
(b)
yp = Ae2t + Bt sin(2t) + Ct cos(2t)
(c)
yp = Ate2t + B sin(2t) + C cos(2t)
(d)
yp = Ate2t + Bt sin(2t) + Ct cos(2t)
3. It is easy to see that the homogeneous solution to y 00 + 2y 0 − 3y = 0 is
yh = C1 et + C2 e−3t . Find a modified guess yp for y 00 + 2y 0 − 3y = e−3t + cos(2t)
(a)
yp = Ate−3t + B cos(2t)
(b)
yp = Ae−3t + B cos(2t) + C sin(2t)
(c)
yp = Ate−3t + B cos(2t) + C sin(2t)
(d)
yp = Ae−3t + Bt cos(2t) + Ct sin(2t)
5
Homogeneous Solutions
1. Solve
(a)
(b)
(c)
(d)

 .25u00 + 4u0 + 7u = 0
u(0) = 0
 0
u (0) = 2
1
u(t) = e−2t −
6
1
u(t) = e−2t +
6
2
u(t) = e−4t −
3
2
u(t) = e−4t −
3
1 −14t
e
6
1 −14t
e
6
2 −7t
e
3
2 −7t
e
3
2. Solve
(a)
(b)
(c)
(d)
y 00 + 5y 0 + 7y = 0
y(0) = 1
y 0 (0) = 0
√
5
5
+ √ e− 2 t sin( 23 t)
3
5
y = e−5t cos(2t) + e−5t sin(2t)
2
√
√
5
5
5
y = e− 2 t cos( 23 t) − √ e− 2 t sin( 23 t)
3
5
y = e−5t cos(2t) − e−5t sin(2t)
2
5
y = e− 2 t cos(
√
3
t)
2
3. Solve
y 00 + 4y 0 + 4y = 0
y(0) = 1
y 0 (0) = 0
(a)
y = e−2t − 2te−2t
(b)
y = e−2t + 2te−2t
(c)
y = e−2t + 2e−4t
(d)
y = e−2t − 2e−4t
6
Spring-mass vibration
1. Each of the following equations describes a spring-mass vibration. Which one of them is
undergo resonance?
(a)
10u00 + 160u = 10 cos(3t)
(b)
10u00 + 160u = 10 sin(3t)
(c)
10u00 + 160u = 4 cos(2t)
(d)
10u00 + 160u = 10 cos(4t)
2. Each of the following equations describes a spring-mass motion. Which one of them is
damped?
(a)
10u00 + 160u = 10 cos(3t)
(b)
10u00 + 160u = 10 sin(3t)
(c)
10u00 + 3u0 + 160u = 4 cos(2t)
(d)
10u00 + 160u = 10 cos(4t)
3. Each of the following equations describes a damped spring-mass motion. Which one of them
is under-damped?
(a)
u00 + 10u0 + 160u = 10 cos(3t)
(b)
u00 + 30u0 + 160u = 10 sin(3t)
(c)
u00 + 8u0 + 16u = 4 cos(2t)
(d)
10u00 + 80u0 + 160u = 10 cos(4t)
4. Each of the following equations describes a damped spring-mass motion. Which one of them
is over-damped?
(a)
u00 + 10u0 + 160u = 10 cos(3t)
(b)
u00 + 30u0 + 160u = 10 sin(3t)
(c)
u00 + 8u0 + 16u = 4 cos(2t)
(d)
10u00 + 80u0 + 160u = 10 cos(4t)
7
5. Each of the following equations describes a damped spring-mass motion. Which one of them
is critically-damped?
(a)
u00 + 10u0 + 160u = 10 cos(3t)
(b)
u00 + 30u0 + 160u = 10 sin(3t)
(c)
u00 + 8u0 + 160u = 4 cos(2t)
(d)
10u00 + 80u0 + 160u = 10 cos(4t)
6. Each of the following equations describes a damped spring-mass motion. Which one of them
is oscillating forever? Check ALL that apply.
(a)
u00 + 10u0 + 160u = 0,
y(0) = 1,
y 0 (0) = .5
(b)
u00 + 30u0 + 160u = 0,
y(0) = 1,
y 0 (0) = .5
(c)
u00 + 16u = 0,
(d)
10u00 + 80u0 + 160u = 0,
y(0) = 1,
y 0 (0) = .5
y(0) = 1,
y 0 (0) = .5
7. u(t) = cos(2t) + sin(2t) models a spring-mass free motion. What is the frequency of the
system?
(a)
2
(b)
3
(c)
5
(d)
It can not be determined without the differential equation.
8. u(t) = cos(2t) + sin(2t) models a spring-mass free motion. What is the amplitude of the
vibration?
(b)
√
3
√
2
(c)
2
(d)
1
(a)
9. u(t) = − cos(2t) + sin(2t) models a spring-mass free motion. What is the phase of the
vibration?
(a)
.785 Rad
(b)
−.785 Rad
(c)
2.355 Rad
(d)
0
8
10. u(t) = e−3t cos(2t) + 3e−3t sin(2t) models a spring-mass free motion. What is the quasifrequency of the system?
(a)
2
(b)
3
(c)
5
(d)
It can not be determined without the differential equation.
Laplace transform
1. Find the inverse Laplace transform of Y (s) =
(a)
(b)
(c)
(d)
1
cos(3t) − sin(3t)
9
1 1
1
y(t) = − cos(3t) − sin(3t)
9 9
3
1 1
y(t) = − cos(3t) + sin(3t)
9 9
1 1
1
y(t) = − cos(3t) + sin(3t)
9 9
3
y(t) =
2. Find the inverse Laplace transform of Y (s) =
(a)
(b)
(c)
(d)
s+1
s3 + 9s
s2
s+5
e−3πs
+ 2s + 10
5 −(t−3π)
−(t−3π)
y(t) = e
cos(3(t − 3π)) + e
sin(3(t − 3π)) u3π (t)
3
4 −(t−3π)
−(t−3π)
sin(3(t − 3π)) u3π (t)
y(t) = e
cos(3(t − 3π)) + e
3
5
y(t) = e−t cos(3t) + e−t sin(3t) e−3πs
3
4 −t
−t
y(t) = e cos(3t) + e sin(3t) u3πs
3
3. Find the Laplace transform of f (t) = (t − π)uπ (t)
(a)
(b)
(c)
e−πs
s2
−πs
e
F (s) =
s
uπ (t)
F (s) = 2
s
F (s) =
9
(d)
F (s) =
2e−πs
s2
4. Solve for Y (s) in the following differential equation.
y 00 + 4y = δ(t − π)
y(0) = 0
y 0 (0) = 1
(a)
(b)
(c)
(d)
e−πs
1
+
s2 + 4 s2 + 4
e−πs
s
+ 2
Y (s) = 2
s +4 s +4
se−πs
Y (s) = 2
s +4
s
Y (s) = 2
s +4
Y (s) =
10