Math 287 Sample Exam Questions for Chapters 4 and 8
spring 2008
1. For each differential equations perform the following
Find the characteristic equation, Find the characteristic roots, Find the general solution to the differential equation.
(a) (b)
413 0xx x−+ =
096
=
′
+
′
′
−
′
′
′
yyy
(c) (d)
0294 =+
′
+
′′ yyy 02 2
2
3
3
4
4=+− dt
yd
dt
yd
dt
yd
2. Let
L
be a second order linear differential operator with constant coefficients.
Suppose and are solutions to
1
y2
y0)(
=
yL , which is a 2nd order linear homogeneous equation and
Suppose
A
ϕ
and B
ϕ
are solutions to , which is a 2
x
eyL sin
)( =nd order linear nonhomogeneous equation
Which of the following are true and which are false and which do you not have enough information to determine?
(a) is a solution to the equation
21 yy +0)(
=
yL
(b) A
y
ϕ
+
1 is a solution to the equation 0)(
=
yL
(c) If and are linearly independent, then
1
y2
y2211 ycycy
+
=
is the general solution to .
0)( =yL
(d) A
y
ϕ
+
1 is a solution to the equation
x
eyL sin
)( =
(e) BA
ϕ
ϕ
+ is a solution to the equation
x
eyL sin
)( =
(f) If
A
ϕ
and B
ϕ
are linearly independent, then 2211
ϕ
ϕ
ccy
+
=
is the general solution to .
x
eyL sin
)( =
3. Find the general solution of the differential equation 2
2t
e
yyy t
=+
′
−
′′ using variation of parameters
4. Find the general solution of the differential equation using undermined coefficients
teyyy t932 2+=−
′
−
′′
5. The charge on a capacitor, , in an LRC circuit is governed by
)(tQ )(
1tvQQRQL C=++ . For a certain LRC
Circuit the charge on the capacitor is given by
()
)3sin(1.7)3cos(1.6)5.3sin(2.12)5.3cos(1.8)( 5.0 ttttetQ t+++−= −
(a) Determine the part of the solution that is transient.
(b) Is the circuit underdamped, critically damped, or overdamped?
(c) Determine the amplitude of the steady state solution
(d) The steady state solution produces a trajectory in the phase plane that is the same curve as the trajectory of an LC
circuit (NO resistance). What is a differential equation that would govern this LC circuit?
6. An object of mass 2 kg is attached to a spring with spring constant 8 kg/m. The object is pulled down 1 meter below its
equilibrium position and given an upward velocity of 2 meters per second.
(a) Write down the
initial value problem whose solution will be give the resulting motion of the object relative to its
equilibrium position. Use the orientation presented in the text and in class; a displacement below the equilibrium position
is a positive displacement and above is a negative displacement. DO NOT solve the IVP
(b) Now, suppose the motion of the object is also affected by a damping force equal to two times the velocity of the
object. Would the resulting system be overdamped, underdamped, or critically damped? Show the work that supports
your answer.
(c) Now, suppose the motion of the object is damped as in (b) and is additionally affected by external force, given by
the function newtons, where represents the number of seconds since the object has been released.
Write down the
ttf cos2)( =t
initial value problem whose solution will be give the motion of the object. DO NOT solve the IVP
(d) Match the phase plane trajectories below with the three cases above. Note one phase plane will have no match
