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Final Review.pdfMath 20D Final Review 1. Solve
Final_Review.pdfMath 20D Final Review 1. Solve
Final Review.pdfMath 20D Final Rev...
Final_Review.pdfMath 20D Final Review 1. Solve
Page 1
Math 20D Final Review
1.
Solve the differential equation in two ways, ﬁrst using variation of parameters and
then using undetermined coefﬁcients:
y
00

5
y
0
+
6
y
=
2
e
t
2.
Draw the direction ﬁeld for
y
0
=
3

2
y
3.
Solve the IVP:
y
00
+
3
y
0
=
0
y
(
0
)=

2;
y
0
(
0
)=
3
4.
Find the general solution for the system of differential equations:
~
x
0
=
✓
1

1
3

2
◆
~
x
+
✓
1

1
◆
e
t
5.
For the given ODE:
(a) Sketch the graph of
f
(
y
)
vs.
y
(b) Determine the critical (equilibrium) points
(c) Classify the equilibrium points
(d) Draw the phase line
(e) Sketch several graphs of solutions in the
ty
plane
dy
dt
=
y
(
y

1
)(
y

2
)
,
y
0
≥
0
6.
Find the Laplace transform of each of the following functions:
(a)
f
(
t
)=
t
(b)
f
(
t
)=
t
2
(c)
f
(
t
)=
t
n
for
n
a positive integer
7.
Find the general solution to the given system of differential equations:
~
x
0
=
0
@
3
2
4
2
0
2
4
2
3
1
A
~
x
1
Page 2
Math 20D Final Review
8.
(a) Seek power series solutions to the diff eq about the given point
x
0
(b) Find the recurrence relation between coefﬁcients
(c) Find the ﬁrst few terms in each of two solutions
y
1
,
y
2
(d) Evaluate the Wronskian
W
(
y
1
,
y
2
)(
x
0
)
. Check that it is indeed nonzero.
y
00

xy
0

y
0
=
0
x
0
=
1
9.
Consider the ﬁrst order system of equations
~
x
0
=
1
5
2
4
~
x
(a) Find a fundamental set of solutions to this system, and VERIFY that your con
structed functions are fundamental.
(b) Sketch the phase portrait of the system, including a few trajectories, and classify
the equilibrium at the origin.
10.
Solve the initial value problem:
(
6
y
2

x
2
+
3
)
dy
dx

2
xy
=

3
x
2

2
y
(
0
)=
1
11.
Find the general solution:
9
y
00
+
9
y
0

4
y
=
0
12.
Solve the system of equations, or show that there is no solution:
x
1
+
2
x
2

x
3
=
2
2
x
1
+
x
2
+
x
3
=
1
x
1
+
2
x
3
=

1
+
x
2
13.
Consider a population
p
of pandas that grows at a rate proportional to the current
population. That is,
dp
dt
=
rp
.
(a) Find the rate constant
r
if the population doubles in 30 days.
(b) Find
r
if the population doubles in
N
days.
2
Page 3
Math 20D Final Review
14.
Solve the IVP:
y
00

2
y
0

3
y
=
3
te
2
t
y
(
0
)=
1,
y
0
(
0
)=
0
15.
Transform the given equations into systems of ﬁrst order equations.
(a)
u
00
+
1
2
u
0
+
2
u
=
0
(b)
u
00
+
1
2
u
0
+
2
u
=
3 sin
t
(c)
u
(
4
)

u
=
0
16.
Find the general solution to the given differential equation.
How do solutions
behave as
t
!
•
?
y
0
+
3
y
=
t
+
e

2
t
17.
Find the general solution to the given system of equations:
~
x
0
=
2
4
1
1
1
2
1

1
0

1
1
3
5
~
x
Hint: The characteristic polynomial is
(
l
+
1
)(
l

2
)
2
=
0.
18.
Find the general solution:
16
y
00
+
24
y
0
+
9
y
=
0
19.
Find the eigenvalues and eigenvectors:
✓
3

2
4

1
◆
20.
Solve the ODE:
y
0
+
y
2
sin
x
=
0
21.
Determine the
a
n
so that the equation
•
Â
n
=
1
na
n
x
n

1
+
2
•
Â
n
=
0
a
n
x
n
is satisﬁed. Try to identify the function represented by
Â
•
n
=
0
a
n
x
n
.
22.
Determine the longest interval in which this IVP is certain to have a unique solution.
Don’t try to solve the system.
t
(
t

4
)
y
00
+
3
ty
0
+
4
y
=
2;
y
(
3
)=
0
y
0
(
3
)=
1
3
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