Intro to Differential Equations Final Exam Review
MATH 2070 – Dr. Lori Alvin
The final exam is cumulative and will cover all the material we discussed this quarter. For
some of the problems I will tell you what technique to use to solve the problem; you will not
receive full credit if you do not use the listed technique.
The following sections are the MOST IMPORTANT to review for the final; however questions
may be taken from any section we discussed this quarter:
1.2: Separation of Variables.
1.9: Integrating Factors for linear equations.
2.3: Damped Harmonic Oscillator don’t forget this factoring technique for solving them.
3.2:
3.3:
3.4:
3.5:
3.6:
Straight-line solutions.
Phase Portraits for Linear Systems with Real Eigenvalues.
Complex Eigenvalues.
Repeated and Zero Eigenvalues.
Second-order Linear Equations.
4.1: Forced Harmonic Oscillators.
4.2: Sinusoidal Forcing.
6.1:
6.2:
6.3:
6.4:
Laplace Transforms.
Discontinuous Functions.
Second-order Equations.
Delta Functions and Impulse Forcing.
Practice Problems:
1.2: Solve the given initial value problem.
dy
= t2 y 3 , y(0) = −1
dt
dy
(b)
= 2y + 1, y(0) = 3
dt
(a)
(c)
dy
= 2ty 2 + 3t2 y 2 ,
dt
(c)
dy 3
− y = 2t3 e2t ,
dt
t
y(1) = −1
1.9: Solve the given initial value problem.
dy
y
= − + 2, y(1) = 3
dt
t
dy
2
(b)
= −2ty + 4e−t , y(0) = 3
dt
(a)
2.3: Use the guess-and-test method to find the general solution.
d2 y
dy
+ y + 10y = 0
2
dt
dt
2
dy
dy
+4 +y =0
(b)
2
dt
dt
(a)
y(1) = 0
3.2/3.3: Compute the general solution and sketch the phase portrait.
dx
dy
dY
−5 −2
(a)
= 5x + 4y,
= 9x
(b)
=
Y
−1 −4
dt
dt
dt
3.4: Solve the initial value problem, determine the type of the system (center, spiral sink,
spiral source), determine the direction of oscillations in the phase plane.
dY
dY
−3 −5
0 2
(a)
=
Y, Y (0) = (4, 0) (c)
=
Y, Y (0) = (1, 0)
3
1
−2 0
dt
dt
dY
1 4
(b)
=
Y, Y (0) = (1, −1)
−3 2
dt
3.5: Solve the initial value problem and sketch the phase plane.
dY
dY
−3 0
0 −2
(a)
=
Y, Y (0) = (1, 0) (c)
=
Y,
1 −3
0 0
dt
dt
dY
2 4
=
Y, Y (0) = (1, 0)
(b)
3 6
dt
Y (0) = (1, 1)
3.6: Write the corresponding first order system, find the eigenvalues, classify the oscillator,
determine the direction of oscillation (if applicable).
dy
d2 y
+3 +y =0
2
dt
dt
2
dy
dy
(b) 9 2 + 6 + y = 0
dt
dt
(a) 2
d2 y dy
+
+ 3y = 0
dt2
dt
d2 y
dy
(d)
+ 6 + 8y = 0
2
dt
dt
(c) 2
4.1/4.2: Solve the initial value problem for y(0) = y 0 (0) = 0
d2 y
dt2
d2 y
(b)
dt2
d2 y
(c)
dt2
d2 y
(d)
dt2
(a)
+4
dy
+ 3y = e−4t
dt
+ 9y = e−t
+ 4y = −3
+6
dy
+ 8y = 2
dt
d2 y
dt2
d2 y
(f)
dt2
d2 y
(g)
dt2
d2 y
(h)
dt2
(e)
dy
+ 20y = −3 sin(2t)
dt
dy
+ 4 + 20y = − cos(5t)
dt
dy
+ 6 + 8y = −4 cos(3t)
dt
+4
+ 4y = 3 sin(2t)
6.1 /6.2: Solve the initial value problem.
dy
dy
(c)
= −y + 2u3 (t), y(0) = 4
= u2 (t), y(0) = 3
dt
dt
dy
dy
(b)
= −y + u1 (t)(t − 1), y(0) = 2 (d)
+ 9y = 2, y(0) = −2
dt
dt
(a)
6.3: Repeat the problems from 4.1/4.2 using Laplace transforms.
d2 y
dy
+ 2 + 5y = δ3 (t), y(0) = 1, y 0 (0) = 1
2
dt
dt
d2 y
dy
(b)
+ 2 + 2y = −2δ2 (t), y(0) = 2, y 0 (0) = 0
dt2
dt
2
dy
(c)
+ 3y = 5δ2 (t), y(0) = 0, y 0 (0) = 0
2
dt
6.4: (a)