MATH 20D Practice Final Problems

Roger Nguyen
Spring 2021

Author’s Note: These problems cover chapters 1.1-1.2, 2.2-2.4, 4.2-4.7, 9.1, 9.3-
9.8, 7.2-7.6, 7.9, and 8.2-8.3 from Fundamentals of Differential Equations (Nagle, Saff,
Snider) 9th Edition. 1.3 is on the final, but I didn’t want to include a direction field
question. Most of these problems are taken from the textbook and lecture examples.
There’s probably some errors in the solution, so keep that in mind.

1 Short Answer Questions

1.2
Is this differerential equation exact, linear, separable, none, or a combination of the
three?
dy
+ y3 = 0
dx

1.3
Is f (x) = x2 − x−1 a solution to the differential equation?

d2 y 2
2
− 2y = 0
dx x

1.4
For the initial value problem, is there a unique solution?
dy 2
= 3y 3 , y(2) = 0
dx

1
1.5
For the initial value problem, is there a unique solution?
dy
x + y = 0, y(0) = 2
dx

1.6
For the initial value problem, is there a unique solution?
y 00 + y = 0, y(0) = 0, y 0 (0) = 0

1.7
For the initial value problem, is there a unique solution?
y 00 + y 0 = 2x, y(1) = 3, y 0 (1) = 1

1.8
For the initial value problem, is there a unique solution?
   
1 0 0 t
0
x (t) = 0 1 0 x(t) + tan(t) , x(0) = 0
   
0 0 1 cot(t)

1.9
Are these two functions linearly independent?
y1 = et
y2 = e2t

1.10
Are these two functions linearly independent?
y1 = 1
y2 = 2

1.11
Write this differential equation in standard form
y 0 + 2y = 2

2
1.12
Write this differential equation in standard form

xy 00 + 4y 0 + 7y = 0

1.13
Express this differential equation as a first order matrix equation

y 00 + y = 0

1.14
Write this system in normal form
    
x0 0 1 0 0 x1
 10  
x −3 0 1 0 x2 
 
 2 
 0 = 
x3   0 0 0 1 x3 
 
x04 2 0 −2 0 x4

1.15
Inverse this matrix " #
1 2
1 3

1.16
Transpose this matrix " #
1 2
1 3

1.17
True or False?
(AB)−1 = A−1 B−1

1.18
" # " #
1 t 1 0
A= t and B = Find
e 0 0 1

d
[AB]
dt

3
1.19
Is this a fundamental solution set?
" #
1
x1 = e2t
2
" #
−2
x2 = e2t
4

1.20
Find the eigenvalues " #
7 7
0 6

1.21
The eigenvalues are λ = 1, 2. Are the eigenvectors that correspond to the eigenvalues
linearly independent?

1.22
True or false: A has distinct eigenvectors? (Hint: is it symmetric?)
 
1 −2 2
−2 1 2
 
2 2 1

1.23
Find eAt where A is " #
3 0
0 1

1.24
True or false: eAt is not a fundamental matrix

1.25
" #
et 0
Given eAt = , find the general solution x(t).
0 et

4
1.26
# "
et 0
Given the fundamental matrix X(t) = , find the eAt
0 1

1.27
Find L {t2 et }

1.28
For what interval of s, does the laplace transform exist for L {5t2 e−3t − e12t cos(8t)}

1.29
Write the function in terms of window and step functions



 3, t < 2,

1, 2 < t < 5,
f (t) =


 t, 5 < t < 8,
2
t ,

8<t
10

1.30
Find L {(t − 1)u(t − 1)}

1.31
Find L {tu(t − 1)}

1.32
−2s
Find L −1 { e s2 }

1.33
Write tΠ2,5 in terms of step functions

1.34
Evaluate Z ∞
(t2 − 1)δ(t)dt
−∞

5
1.35
Evaluate Z ∞
e−2t δ(t + 1)dt
−∞

1.36
Find L {δ(t − 2)}

1.37
Find L {sin(t)δ(t − π)}

1.38
Determine the interval of convergence of
∞
X (−2)n
(x − 3)n
n=0
n+1

1.39
Write out the power series for
1
1−x

1.40
Determine all the singular points of
x 0
xy 00 + y + (sinx)y = 0
1−x

1.41
Is the power series below a solution to y 0 + 2xy = 0?
∞
X 1
y= (−x2 )n
n=0
n!

(Hint: Write the power series as a function)

6
2 Differential Equations
Author’s Note: Any method is fair game. You could use auxillary polynomial for
most second order linear DEs with IVPs but some require Laplace.

2.1
Find a general solution
dy
= y2 − 1
dx

2.2
Find a general solution  
dy
x 2 + 1 = 3y
dx

2.3
Find a general solution
dy
−(x2 + 2y) = 2xy − sec2 x
dx

2.4
Find a general solution
y 000 − 3y 00 + 4y 0 = 12y

2.5
Find a general solution
y 00 + 4y 0 + 4y = 0

2.6
Find a general solution
y 00 + 2y 0 + 4y = 0

2.7
Find a particular solution
y 00 + y = t2 + 1

7
2.8
Find a particular solution
y 00 − y = 8tet + 2et

2.9
Find a particular solution
y 00 − 4y 0 − 12y = sin(2t)

2.10
Find a general solution
y 00 + y = tan(t)

2.11
Find a general solution
3t2 y 00 + 11ty 0 − 3y = 0

2.12
Find a general solution
t2 y 00 + 5ty 0 + 5y = 0

2.13
Find a general solution
t2 y 00 + ty 0 = 0

2.14
Given y1 (t) = t is a solution to

t2 y 00 − ty 0 + y = 0

Find a second linearly independent solution

2.15
Find a general solution "#
0 1 2
x (t) = x(t)
3 2

8
2.16
Find a general solution " #
−1 2
x0 (t) = x(t)
−1 −3

2.17
Find a particular solution
" # " #
1 2 t
x0 (t) = x(t) +
3 4 2t

2.18
Find a particular solution
   
2t
0 −1 0 e
x0 (t) = −1 0 0 x(t) + sin(t)
   
0 0 1 t

2.19
Find a general solution to the nonhomogenous system given the fundamental matrix
" #
e−t −e−6t
X(t) =
4e−t e−6t
" #
2t
6e
and f (t) =
−e2t

2.20
Find the fundamental matrix eAt for the system
 
2 1 1
x0 (t) =  1 2 1  x(t)
 
−2 −2 −1

(Hint: Use the Cayley-Hamilton Theorem)

9
2.21
Find the fundamental matrix eAt for the system
 
1 0 0
x0 (t) = 1 3 0 x(t)
 
0 1 1

2.22
Solve the IVP given y(0) = 2 and y 0 (0) = 12
y 00 − 2y 0 + 5y = −8e−t

2.23
Solve the IVP given y(π) = 2 and y 0 (π) = 12
y 00 − 2y 0 + 5y = −8eπ−t

2.24
Solve the IVP given y(0) = 0 and y 0 (0) = 0
y 00 + 2ty 0 − 4y = 1

2.25
Solve the IVP given y(0) = 2 and y 0 (0) = −4
ty 00 − ty 0 + y = 2

2.26
Solve the IVP given y(0) = 0 and y 0 (0) = 0
y 00 + 4y = g(t)

1, 0 < t < 1


g(t) = −1, 1 < t < 2

0, 2 < t


2.27
Solve the IVP given y(0) = 1 and y 0 (0) = 0
y 00 + 9y = 3δ(t − π)

10
2.28
Solve the DE
y 0 + 6x2 y = 0
(Hint: answer should not be in the form of a power series)

2.29
Find the sum up to the a3 term in a power series expansion to the DE in terms of a0
and a1 .
(1 + x2 )y 00 − y 0 + y = 0

11
3 Solutions
Problem Solution Problem Solution
1.1 exact and separable 1.20 λ = 7, 6
1.2 separable 1.21 yes
1.3 yes 1.22 true
" #
3t
e 0
1.4 no 1.23
0 et
1.5 no 1.24 false
" # " #
et 0
1.6 yes 1.25 x(t) = c1 + c2 t
0 e
" #
et 0
1.7 yes 1.26
0 1
2
1.8 no 1.27 (s−1)3

1.9 yes 1.28 (12, ∞) or s > 12

y 00 + x4 y 0 + x7 y −s 1 1


1.12 =0 1.31 e s2
+ s
" # " #" #
x01 0 1 x2
1.13 0 = 1.32 (t − 2)u(t − 2)
x2 −1 0 −x1
x01


1.14 = x2 1.33 t u(t − 2) − u(t − 5)
x02 = −3x1 + x3 1.34 −1
x03 = x4 1.35 e2
x04 = 2x1 − 2x3 1.36 e−2s
" #
3 −2
1.15 1.37 0
−1 1
" #
1 1
1.16 1.38 ( 25 , 72 )
2 3
P∞
1.17 false 1.39 n=0 xn
" #
0 1
1.18 1.40 x=1
et 0
1.19 yes 1.41 yes

12
P roblem Solution
  
1  y−1 
 2 ln y+1  = x + C


2.1 y = −1


y = 1
3
2.2 y = x + Cx 2
2.3 C = x2 y − tan(x) + y 2
2.4 y = C1 e3t + C2 cos(2t) + C3 sin(2t)
2.5 y = C1 e−2t + C2 te−2t
√ √
2.6 y = C1 e−t cos( 3t) + C2 e−t sin( 3t)
2.7 yp = t2 − 1
2.8 yp = (2t2 − t)et
1 1
2.9 yp = 40
cos(2t) − 20
sin(2t)
2.10 y = C1 cost + C2 sint − (cost) ln|sect + tant|
2.11 y = C1 t1/3 + C2 t−3
2.12 y = C1 t−2 cos(ln t) + C2 t−2 sin(ln t)
2.13 y = C1 + C2 ln t
2.14 y2 = t ln t
"# " #
−1 2
2.15 x(t) = C1 e−t + C2 e4t
1 3
" # " #
2cos(t) 2sin(t)
2.16 x(t) = C1 e−2t + C2 e−2t
−(cos(t) + sin(t)) cos(t) − sin(t)
" # " #
0 −1/2
2.17 xp = t+
−1/2 1/4

13
P roblem Solution
 
2 2t
e + 1 sint
 3 1 2t 2 1
2.18 xp = − 3 e − 2 cost

−t − 1
" #
23/24
2.19 xp = e2t
17/24
    
1 0 0  1 1 1
 
2.20 eAt = et 
  0 1 0 +  1 1 1  t
0 0 1 −2 −2 −2
 
t
e 0 0
2.21 eAt =  − 12 et + 12 e3t e3t 0
 
− 14 et − 12 tet + 41 e3t − 12 et + 12 e3t et
2.22 y = 3et cos(2t) + 4et sin(2t) − e−t
2.23 y = 3et−π cos(2t) + 4et−π sin(2t) − eπ−t
2.24 y = 12 t2
2.25 y = 2 − 4t
y = 14 − 14 cos(2t) − 21 − 12 cos(2t − 2) u(t − 1) + 41 − 41 cos(2t − 4) u(t − 2)

   
2.26
2.27 y = cos(3t) + [sin(3t − 3π)]u(t − π)
3
2.28 y = C1 e−2x
y = a0 1 − 12 x2 − 61 x3 + a1 x + 21 x2



2.29

14