Problems: Tue 9/27
Sec 1.1
2. Consider the ODE given by y 0 = 2y − 3.
(a) Draw a direction field.
(b) Determine the behavior of y(t) as t → ∞. Does your answer depend
on the initial value y(0)?
(c) Determine the equilibrium solutions. Which are stable / unstable /
semi-stable?
12. Same question for y 0 = −y(5 − y).
14. Same question for y 0 = y(y − 2)2 .
21. Same question for y 0 = −2 + t − y.
30. Verify that the function u(t) = T0 + ce−kt is a solution of
du
+ ku = kT0 .
dt
Problems: Thu 9/29
Sec 1.2
14. Solve the initial-value problem
y 0 + 2y = te−2t
y(1) = 0.
16. Solve the initial-value problem
2
cos t
y= 2 ,
t
t
y(π) = 0.
y0 +
t>0
31. Consider the initial-value problem
y 0 − 32 y = 3t + 2et
y(0) = y0 .
Find the value of y0 that separates solutions that grow positively as t → ∞
from those that grow negatively. How does the solution that corresponds to
this critical value of y0 behave as t → ∞?
33. Let a, λ > 0 be positive constants, and let b ∈ R be a real number.
Show that every solution of the equation
y 0 + ay = be−λt
has the property that y → 0 as t → ∞.
Be careful not to divide by zero.
38. Consider the initial-value problem
y 0 + ay = g(t)
y(t0 ) = y0 .
Assume that a > 0 is a positive constant, and that lim g(t) = g0 .
t→∞
g0
Show that lim y(t) = .
t→∞
a
Problems: Tue 10/4
#1-5. For each matrix (or linear transformation):
(a) Find the eigenvalues. What are their algebraic multiplicities?
(b) Find the eigenvectors. What are the geometric multiplicities of each
eigenvalue? Draw the eigenspaces corresponding to real eigenvalues (if any).
5 −1
1.
3 1
3 −2
2.
4 −1
1 k
3.
0 1


1 1 0
4. 0 1 1
0 0 1
5. The linear transformation T : R2 → R2 given by projection onto the line
y = 2x.
Do this problem without writing down any matrices.
Drawing pictures is encouraged.
6. Consider the matrix


1 0 0
A = 2 1 −2
3 2 1
One of the eigenvalues is λ1 = 1 + 2i.
(a) Find the other two eigenvalues λ2 , λ3 without doing any serious calculations.
(b) Show that (0, i, 1) is an eigenvector of A. What is its corresponding
eigenvalue?
Problems: Thu 10/6
#1-4. For each matrix (or linear transformation):
(a) Find the eigenvalues. What are their algebraic multiplicities?
(b) Find the eigenvectors. What are the geometric multiplicities of each
eigenvalue? Draw the eigenspaces corresponding to real eigenvalues (if any).


7 1 0
1. 0 7 1
0 0 7


3 0 0
2. 0 3 1
0 0 3
3. The linear transformation T : R3 → R3 given by reflection in the plane
z = 0.
Do this problem without writing down any matrices.
4. The linear transformation S : R5 → R5 given by dilation by a factor of 3.
Do this problem without writing down any matrices.
5. Find all values of the following expressions:
(a) (8i)1/3
(b) (−16)1/4
(c) Ln(−2i).
0 −π
6. Let A =
. Find eA .
π 0
(You will need to find the eigenvalues of A, the eigenvectors of A, and then
write A = T DT −1 with D diagonal.)
Recall: Let z = reiθ . Then:
θ
z 1/n = (reiθ )1/n = r1/n e( n +
2πk
n )i
,
k = 0, . . . , n − 1.
We also have
Ln(z) = Ln(r) + i(θ + 2πk),
k ∈ Z.
Problems: Tue 10/11
#1-4. For each system x0 = Ax:
(a) Draw a direction field and phase portrait.
(b) Find the eigenvalues and eigenvectors of A.
(c) Find the general solution.
(d) Draw the component plots.
(e) Give an explicit formula for the phase curves.
2 0
0
x.
1. x =
0 8
−2 0
0
2. x =
x.
0 −8
−2
0
3. x0 =
x.
0 8
0
−1
4. x0 =
x.
1 0
Problems: Thu 10/13
#1-3. For each system x0 = Ax:
(a) Find the eigenvalues and eigenvectors of A.
(b) Find the general solution and a fundamental matrix.
(c) Draw a phase portrait.
(d) Give an explicit formula for a phase curve that does not involve t.
5
1
1. x0 =
x.
0 5
0
−1
2. x0 =
x.
1 0
2
−1
3. x0 =
x.
1 2
a b
4. Let A =
. Let T = tr(A) and D = det(A).
c d
Prove that the characteristic polynomial of A is
p(λ) = λ2 − T λ + D.
√
Then show that the eigenvalues are λ = 12 (T ± T 2 − 4D).
−1
−α
5. Consider x0 =
x.
−1 −1
(a) Determine the eigenvalues (in terms of α).
(b) Find the critical values of α where the qualitative nature of the phase
portrait changes.
Problems: Tue 10/18
1 −1
1. Let A =
.
5 −3
1 −1
1
1
= (−1 + i)
.
It is a fact that
5 −3
2−i
2−i
(a) Find a fundamental set of solutions to x0 = Ax.
(b) Use the Wronskian to show that your fundamental set of solutions is,
in fact, a fundamental set of solutions.
2. Consider the ODE system
x0 (t) = x(t) − y(t) + 2
y 0 (t) = 5x(t) − 3y(t) + 2.
(a) Find the equilibrium solution(s), if any.
(b) Classify the equilibrium solution(s).
(c) Find the general solution of the ODE system.
3. Let A be a matrix whose characteristic polynomial is
p(λ) = (λ − 7)3 (λ + 5).
(a) Given only this information, list all possible Jordan forms of A.
(b) Suppose you also know that λ = 7 has geometric multiplicity 2. What
is the Jordan form of A?
4. Find all values of Ln(1 − i).
More Problems
a b
5. Let A =
. Let T = tr(A) and D = det(A).
c d
Prove that the characteristic polynomial of A is
p(λ) = λ2 − T λ + D.
√
Then show that the eigenvalues are λ = 12 (T ± T 2 − 4D).
−1 −α
0
x.
6. Consider x =
−1 −1
(a) Determine the eigenvalues (in terms of α).
(b) Find the critical values of α where the qualitative nature of the phase
portrait changes.
7. In each case,
find
the equilibrium solutions, if any.
1 1
(a) x0 =
x
1 1
1 1
2
0
(b) x =
x+
1 1
3
1 1
5
(c) x0 =
x+
1 1
5
Problems: Thu 10/20
1. Consider the 1st-order linear ODE system
−3 1
1
0
x+
.
(?)
x =
1 −3
4t
−3 1
has eigenvalues and eigenvectors:
Note that the matrix A =
1 −3
1
◦ λ1 = −2 with v1 =
1
1
◦ λ2 = −4 with v2 =
.
−1
Find the general solution to the system (?).
2
4
2. Consider x0 =
x.
α 2
(a) Find the eigenvalues of the matrix (in terms of α).
(b) Find the values of α where the nature of the equilibrium solution
changes.
3. In each case,
find
the equilibrium solutions, if any.
1 1
(a) x0 =
x
1 1
1
1
2
(b) x0 =
x+
1 1
3
1 1
5
(c) x0 =
x+
1 1
5
4. Challenge: Clearly explain how the “method of integrating factors” can
be derived as a special case of the formula for Variation of Parameters.
Problems: Tue 10/25
Sec 2.1
3. Find all solutions of the 1st-order ODE
y 0 + y 2 sin x = 0.
10. Consider the 1st-order initial-value problem (IVP)

y 0 = 1 − 2x
y

y(1) = −2.
(a) Find all solutions of the IVP.
(b) Draw the graph of the solution(s) of the IVP.
(c) Find the interval on which your solution(s) is defined.
26. Consider the 1st-order initial-value problem (IVP)
(
y 0 = 2(1 + x)(1 + y 2 )
y(0) = 0.
(a) Find all solutions of the IVP.
(b) Determine where the solution attains its minimum value.
Sec 2.3
1. Consider the 1st-order initial-value problem (IVP)
(
(t − 3)y 0 + (ln t)y = 2t
y(1) = 2.
Without solving the problem, find an interval on which a solution is certain
to exist.
6. Same question as the previous one, but now for the IVP given by:
(
(ln t)y 0 + y = cot t
y(2) = 3.
8. Consider the IVP
(
y 0 = (1 − t2 − y 2 )1/2
y(t0 ) = y0 .
Find a region R in the ty-plane such that for any choice of initial-value
(t0 , y0 ) ∈ R, there will exist a unique solution to the IVP.
14. Consider the IVP
(
y 0 = 2ty 2
y(0) = y0 .
(a) Solve the IVP.
(b) Explain how the interval on which the solution exists depends on y0 .
Fun Question. Consider the IVP
(
ty 0 + y = 7 + sin(sin(sin(t)))
y(0) = 0.
Without doing any involved calculations, explain why this IVP cannot have
a solution.
Problems: Thu 10/27
Sec 2.3
6. Consider the 1st-order IVP
(
(ln t)y 0 + y = cot t
y(2) = 3.
Without solving the problem, find an interval on which a solution is certain
to exist.
14. Consider the 1st-order IVP
(
y 0 = 2ty 2
y(0) = y0 .
(a) Solve the IVP. (Your answer should be in terms of y0 .)
(b) Explain how the interval on which the solution exists depends on y0 .
Approximation of Solutions
1. Let φ(t) be the solution to the 1st-order IVP


y 0 = 1 y − 6y 3
t
1

y(1) = .
2
(a) Use Euler’s method with step size ∆t = 0.1 to approximate φ(1.2).
(b) Use the Improved Euler Method with step size ∆t = 0.1 to approximate
φ(1.2).
2. Challenge: Explicitly solve the IVP in problem 1.
Problems: Tue 11/1
1st-Order ODE: Bifurcations
1. Consider the 1st-order ODE
y 0 = y 3 + ay.
(a) Find the equilibrium solutions (in terms of a). How many are there?
(b) Sketch the graph of f (y) = y 3 + ay for a < 0, a = 0, and a > 0.
(c) Sketch solution curves to the ODE for a < 0, a = 0, and a > 0. Discuss
stability.
(d) Draw the bifurcation diagram. (i.e.: Sketch the level set y 3 + ay = 0 in
the ay-plane.)
2nd-Order Linear ODE
#2-3. For each of the given 2nd-order linear ODE:
(a) Find the general solution y(t).
(b) Write down the corresponding “dynamical system” (1st-order ODE
system). Find its general solution.
(c) Sketch the phase portrait of the dynamical system.
(d) Discuss lim y(t). Does your answer depend on y(0) and y 0 (0)?
t→∞
2. y 00 + 2y 0 − 3y = 0.
3. y 00 − 2y 0 + y = 0.
4. Consider the 2nd-order IVP
y 00 + 4y 0 + 5y = 0
y(0) = 1
y 0 (0) = 0.
(a) Find the solution y(t).
(b) Sketch the graph of y(t).
Problems: Thu 11/3
Spring-Mass Problems (from Sec 4.4) (Take g ≈ 32 ft/s2 ≈ 9.8 m/s2 )
1. Write y = 3 cos 2t + 4 sin 2t in the form y = R cos(ω0 t − δ).
5. A mass of weight 2 lb stretches a spring 6 in = 21 ft.
Suppose the mass is pulled down an additional 3 in = 14 ft and then released, and suppose there is no damping.
(a) Determine the position y of the mass at any time t.
(b) Find the frequency, period, and amplitude of the motion.
6. A mass of 100 g stretches a spring 5 cm.
Suppose the mass is set in motion from the equilibrium position with a
downward velocity of 0.1 m/s, and suppose there is no damping.
Determine the position y of the mass at any time t.
Misc Problems
M1. Sketch the following level sets in the xy-plane.
(a) y − x3 = 0.
(b) x(y − x3 ) = 0.
(c) x2 (y − x3 ) = 0.
(d) (x2 − 2x)(y 2 − 2y) = 0
(e) y 2 − xy + y sin x − x sin x = 0.
(f) xy = c, for different values of c ∈ R.
M2. Consider the 2nd-order linear ODE
ty 00 + 2y 0 = 3t.
Notice that it is not homogeneous, nor constant-coefficient.
Solve the ODE. (Hint: Make a change-of-variable to transform the given
2nd-order ODE into a single 1st-order ODE.)
M3. Find a 2nd-order ODE of the form y 00 + q(t)y = 0 for which the function
y(t) = exp(t2 ) is a solution.
Problems: Tue 11/8
Undetermined Coefficients
Find the general solution of the following 2nd-order ODEs.
1. y 00 + y = 3 sin 2t
2. y 00 + 2y 0 + y = 2e−t
3. y 00 − 2y 0 − 3y = −3te−t
4. u00 + a2 u = cos bt, where a2 6= b2 .
Happy Challenge Problems
H1. Consider the 4th-order linear ODE
y (4) + 25y 00 = cos t.
Find the general solution.
H2. Consider the 3rd-order linear ODE
y 000 − 2y 00 + y 0 − 2y = 0.
Convert it into a 1st-order linear ODE system (of three variables x1 , x2 , x3 ).
Problems: Thu 11/10
Set 1: Theory
1. (a) Calculate L{eat }.
(b) Calculate L{1}. (There are at least three ways to do this.)
2. Calculate L{sin(at)}.
3. Prove that L{eat f (t)} = F (s − a), where F (s) = L{f (t)}.
√
4. Calculate cos( −1).
Set 2: Examples
5. Find the Laplace transform of:
(a) f (t) = t3 − 4t2 + 5.
(b) f (t) = e3t cos 2t.
6. Let n ∈ Z+ and a ∈ R.
Calculate L{tn eat }. (There are at least two ways to do this.)
Set 3: Application to Linear ODEs
20. Find the Laplace transform Y = L{y} of the solution to the IVP:
y 000 + y 00 + y 0 + y = 0
y(0) = 1
y 0 (0) = 0
y 00 (0) = −2.
(Hint: Transform the ODE into an algebraic equation for Y = L{y}. Then solve the algebraic equation for Y (s).)
17. Same problem as previous, this time for the following IVP:
y 00 − 2y 0 − 3y = t2 + 4
y(0) = 1
y 0 (0) = 0.
Problems: Tue 11/15
1
.
1. Calculate L−1 2
2
s
+
b
1
2. Calculate L−1
.
(s − a)n
Sec 5.3: Inverse Laplace Transforms & Partial Fractions
2
11. Calculate L−1 2
.
s + 3s − 4
2s
+
2
.
13. Calculate L−1 2
s + 2s + 5
2s
−
3
14. Calculate L−1 2
.
s −4
2
8s
−
4s
+
12
16. Calculate L−1
.
s(s2 + 4)
Sec 5.4: Application to ODEs
2. Solve the following 2nd-order linear IVP
y 00 + 3y 0 + 2y = t
y(0) = 1
y 0 (0) = 0.
13. Solve the following 4th-order linear IVP
y (4) − 4y = 0
y(0) = 1
y 0 (0) = 0
y 00 (0) = −2
y 000 (0) = 0.
Problems: Thu 11/17
Conceptual
1. Consider f (x) = |x|.
(a) Find the Taylor series of f (x) centered at x = 1, if it exists.
(b) Find the Taylor series of f (x) centered at x = −1, if it exists.
(c) Find the Taylor series of f (x) centered at x = 0, if it exists.
Practice with Indexing
2. In each of the following, write the given expression as a single
Psumnwhose
n
generic term involves x . (That is, the answer should look like
bn x .)
∞
X
(a) x
n(n − 1)an xn−2
(b) x
n=2
∞
X
n−1
nan x
n=1
(c) (1 − x2 )
∞
X
+
∞
X
ak xk
k=0
n(n − 1)an xn−2
n=2
3. Show that
∞
X
n=1
∞
∞ X
X
1 n+1 1
1
1
1
n
xn .
x +
x
= x+
+
n+1
n
2
n+1 n−1
n=1
n=2
Application to Linear ODE
4. Consider the 2nd-order ODE
y 00 − xy 0 − y = 0.
The goal of this problem is to find a series solution centered at x0 = 0.
(a) Find the recurrence relation.
(b) Solve the recurrence relation. (i.e.: Find a closed form for an .)
(c) Find the series solution to the ODE.
Problems: Tue 11/29
Sec 8.2: Power Series Solutions
7. Consider the 2nd-order ODE
y 00 + xy 0 + 2y = 0.
The goal of this problem is to find a series solution centered at x0 = 0.
(a) Find the recurrence relation.
(b) Solve the recurrence relation. (i.e.: Find a closed form for an .)
(c) Find the series solution to the ODE.
Sec 9.2: Fourier Series
For each of the following, find the Fourier series for the given function.
13. The function having f (x + 2L) = f (x) and
f (x) = −x for x ∈ [−L, L).
16. The function having f (x + 2) = f (x) and
(
x + 1 if x ∈ [−1, 0)
f (x) =
1 − x if x ∈ [0, 1).
20. The function having f (x + 2) = f (x) and
f (x) = x for x ∈ [−1, 1).
21. The function having f (x + 4) = f (x) and
1
f (x) = x2 for x ∈ [−2, 2].
2
Problems: Thu 12/1
Sec 9.2: Fourier Series
For each of the following, find the Fourier series for the given function.
16. The function having f (x + 2) = f (x) and
(
x + 1 if x ∈ [−1, 0)
f (x) =
1 − x if x ∈ [0, 1).
20. The function having f (x + 2) = f (x) and
f (x) = x for x ∈ [−1, 1).
Sec 9.8: The Laplace Equation
1. (a) Find the solution u(x, y) of the Laplace equation uxx + uyy = 0 in the
rectangle (0, a) × (0, b) that satisfies the Dirichlet conditions
u(x, 0) = 0
u(x, b) = g(x)
u(0, y) = 0
u(a, y) = 0.
(b) Find the solution if
g(x) =
(
x
if x ∈ [0, 12 a]
a − x if x ∈ [ 21 a, a].
D5. (a) Find the solution u(x, y) of the Laplace equation uxx +uyy = 0 in the
semi-infinite strip (0, ∞) × (0, π) that satisfies both the Dirichlet conditions
u(x, 0) = 0
u(x, π) = 0
u(0, y) = f (y)
and also the condition that u(x, y) → 0 as x → ∞.
(b) Find the solution if f (y) = y(π − y).
Answers to Thu 12/1
16. See textbook: Sec 9.2: #16.
20. See textbook: Sec 9.2: #20.
1. See textbook: Sec 9.8: #1.
D5. This question is not from the book. (But it is similar to Sec 9.8: #8.)
(a) The solution is
Z
∞
X
2 π
−nx
u(x, y) =
cn e
sin(ny), where cn =
f (y) sin(ny) dy.
π
0
n=1
(b) One finds that

 8
cn = πn3
0
if n odd
if n even.
Therefore, the solution is
∞
X
8
1
u(x, y) =
e−(2n−1)x sin((2n − 1)y)
3
π (2n − 1)
n=1
8 sin(y) sin(3y) sin(5y)
+ 3 3x + 3 5x + · · ·
=
π
ex
3e
5e
(pretty!)
Challenge Problem
Show that there cannot be an Identity Principle for trigonometric series
by doing the following: Find an actual example of a trigonometric series
nπ nπ a0 X
+
an cos
x + bn sin
x
2
L
L
which is equal to the zero function on the interval (0, 1), yet not all of the
coefficients an , bn are zero.
Hint: Consider the Fourier series of the two functions defined by
(
−1 if x ∈ [−1, 0)
f (x + 2) = f (x) and f (x) =
1
if x ∈ [0, 1)
and
g(x) ≡ 1.
Then think about the function h(x) = f (x) − g(x).
Problems: Tue 12/6
Sec 9.8: The Laplace Equation
D5. (a) Find the solution u(x, y) of the Laplace equation uxx +uyy = 0 in the
semi-infinite strip (0, ∞) × (0, π) that satisfies both the Dirichlet conditions
u(x, 0) = 0
u(x, π) = 0
u(0, y) = f (y)
and also the condition that u(x, y) → 0 as x → ∞.
(b) Find the solution if f (y) = y(π − y).
6. Find the solution u(r, θ) of the Laplace equation in the half-disk r < 2,
0 < θ < π that satisfies the Dirichlet boundary conditions
u(r, 0) = 0
u(r, π) = 0
u(2, θ) = θ(π − θ)
and also the condition that u(r, θ) is bounded.