Math 4430 HW 11
Selected Solutions
4.1 p. 377
4. Find all of the equilibrium points of the system
dx
= x xy 2
dt
dy
= y yx2
dt
dz
= 1 z + x2 .
dt
Solution
The equilibrium solutions are the constant solutions to the system
(1)
(2)
(3)
0 = x xy 2
0 = y yx2
HOMEWORK I SOLUTIONS
From Section 1.4 in the book: 3, 4, 6, 8, 16, 18.
Problem 3. Solve the dierential equation
y = 1 t + y2 ty2
Sol. Since
y = 1 t + y2 ty2 = 1 t + (1 t)y2 = (1 t)(1 + y2 ),
the equation is separable. Separating y and t in the above then
TEST 1 - ODE REVIEW
The solutions are at the end. I suggest you try to solve the problems rst and, then, check your
solution. That way you will get some practice in determining what type of equation you are facing
and which method to use. Also note that t
FINAL EXAM REVIEW
1. Determine all singular points of the equation.
a) (2 2)y + 2y + (sin )y = 0.
b) (sin x)y + (cos x)y = 0.
2. Find at least the rst four nonzero terms in a power series expansion about x = 0 for a general
solution to the given equation.
HOMEWORK V SOLUTIONS
1. Determine all singular points of the dierential equation:
a)
(x + 1)y x2 y + 3y = 0.
Sol. Here
3
x2
,
q(x) =
.
x+1
x+1
The singular points of the equation are those points x where either p(x) or q(x) are not analytic.
Hence the onl
MIDTERM 1 SOLUTIONS
1) (20 pts.) Solve the IVP (you should be able to write the solution explicitly).
3x2 + 4x + 2
,
y(0) = 1
2y + 1
State in very general terms the restriction needed for y to remain real-valued. You do not have to specify particular
valu
Common Laplace Transforms and Main Properties
Common Transforms.
(1)
1
L cfw_1 (s) = ,
s
s > 0.
(2)
L eat (s) =
1
,
sa
s > a.
(3)
L cfw_tn (s) =
n!
sn+1
,
n = 1, 2, .,
s > 0.
(4)
L cfw_sin(bt) (s) =
b
,
s 2 + b2
s > 0.
L cfw_cos(bt) (s) =
s
,
s 2 + b2
s
HOMEWORK IV SOLUTIONS
Section 2.4 in the book: 2, 3, 9 (Must use variation of parameters method)
For these three problems we use the variation of parameters formulae derived in class:
g(t)y2 (t)
g(t)y1 (t)
dt,
u2 (t) =
dt
(0.1)
a(t)W(y1 , y2 )
a(t)W(y1 ,
HOMEWORK VII SOLUTIONS
Problems to be graded: 4 from section 2.11, and 5, 7, 13 from section 2.13.You will, however,
receive some credit for work done on the other three problems as long as you made a decent eort
to obtain a solution.
Remark: Throughout t
HOMEWORK VI SOLUTIONS TO ASSIGNED PROBLEMS
Only problem 5 from section 2.9 and problems 9, 21 and 24 from section 2.10 will be graded. You
will, however, receive some credit for work done on the other seven problems as long as you made
a decent eort to ob
Fundamental Matrix Solutions
Now that we can solve the homogeneous equation
y = Ay
we will develop some machinery that will become useful as we move toward techniques for solving
the nonhomogeneous equation.
Recall that the general solution to the homogen
HOMEWORK II SOLUTIONS
From Section 1.9 in the book: 3, 8, 12, 16.
From Section 1.8 in the book: 9.
Problem 3. Find the general solution of
dy
=0
dt
Sol. We check if the equation is exact. Multiplying both sides by dt we get
2t sin y + y3 et + t2 cos y + 3
HOMEWORK III SOLUTIONS
Section 2.2 in the book: 4, 6, 8.
Problem 4. Find a general solution to
3y + 6y + 2y = 0.
Sol. This is a second order, homogeneous, constant coecient linear equation. So we try y = ert
to derive the characteristic equation
3r2 + 6r
Math 4430 HW 12
Selected Solutions
4.4 p. 398
2. Verify that x(t) = ln(1 + t), y(t) = et is a solution of the system
x
y
= ex
x
= ee 1
and nd the corresponding orbit.
Solution Plugging in the solution to our system, equation one becomes
x = ex
1
= e ln(1+
Math 4430 HW 10
Selected Solutions
3.12 pp. 366
4. Use the method of variation of parameters to solve the IVP:
y=
0 1
1 0
f1 (x)
f2 (x)
y+
,
y(0) =
0
0
Solution
We start by nding a fundamental matrix solution to the homogeneous equation. We have
det(A I)
Math 4430 HW 8
Selected Solutions
3.1 pp. 271-272
2. Convert the dierential equation
y + cos y = et
to a system of rst-order equations.
Solution Since this equation is third-order, we make the substitutions
y1 = y
y2 = y
y3 = y .
This gives immediately th
Math 4430 HW 9
Selected Solutions
3.9 pp. 344
4. Find the general solution of the system
1 0 1
y = 0 1 1 y
2 0 1
Solution
To nd the eigenvalues we compute
1
0
1
1
1
det(A I) = det 0
2
0
1
= (1 )2 (1 ) + 2(1 )
= ( 1)(2 + 1).
Setting each of these factors
Math 4430 HW 6
Selected Solutions
2.5 pp. 164
4. Find a particular solution to the dierential equation
y + y + y = 1 + t + t2 .
Solution Since the RHS is a polynomial and c = 1 = 0, the method of judicious guessing
tells us to guess a particular solution
Math 4430 HW 7
Selected Solutions
2.10 pp. 237
4. Find the Laplace transform of the function
t2 cos(at).
Solution
We know that
s
s 2 + a2
By property 1, multiplying by t on the left is the same as dierentiation on the right, so
we have
d
s
a2 s 2
L [t cos
Math 4430 HW 4
Selected Solutions
2.1 pp. 136-137
2. (g) Let L[y](t) = y (t) 6y (t) + 5y(t) and compute L[t2 + 2t].
Solution
We have y(t) = t2 + 2t. We begin by nding y (t) and y (t):
y (t) = 2t + 2
y (t) = 2.
Plugging these into the formula for L[y(t)] w
Math 4430 HW 2
Selected Solutions
1.9 pp. 66-67
6. Find the general solution of the dierential equation
y2
dy
2yet + (y et )
= 0.
2
dt
Solution In this case we have
M=
y2
2yet
2
and
N = y et .
We compute
M
= y 2et
y
and also
N
= et .
t
Since these are n
Math 4430 HW 1
Selected Solutions
1.2 pp. 9-10
8. Find the solution to the initial-value problem
dy
+ 1 + t2 y = 0,
dt
y(0) =
5.
Solution Since our equation is homogeneous, the general solution is given by
y = Ce
where in this case we have
R
f (t) dt
f (t
Math 4430 HW 5
Selected Solutions
2.2 pp. 149-150
2. Find the general solution of the dierential equation
4y 12y + 9y = 0.
Solution
We compute b2 4ac = (12)2 4(4)(9) = 0. Then the characteristic equation has a double
root
r = b/(2a) = 12/8 = 3/2.
Then the
TEST 2 - ODE REVIEW
For the sake of comparison, solutions to the problems can be found at the end. I suggest you try
and solve them rst, then check your nal answer. Also note that the same solution can be written
in several dierent ways. So if your soluti
The Eigenvector/Eigenvalue Method
We rst recall the following from linear algebra:
Definition Let A be an n n matrix. A vector v such that
Av = v
for some scalar is called an eigenvector (EV) of A with eigenvalue (ev) .
Before we discuss how these objects
Complex Eigenvalues
Background on Complex Eigenvectors/Eigenvalues
Recall that given a matrix A, we said that a nonzero vector v was an eigenvector of A with eigenvalue
if
Av = v
where is a scalar. Nothing about this denition required that be a real numb
Linear Eqs With Constant Coefficients
We continue to look at the dierential equation
ay + by + cy = 0
but now consider the case where the roots of the characteristic equation are complex. We rst build
some tools to help us toward this goal.
Complex-valued
Algebraic Properties of Solutions
2nd-Order Linear Homogeneous
We will begin our study of second-order dierential equations with the linear homogeneous case,
that is we will consider dierential equations of the form
y + p(x)y + q (x)y = 0.
Since this equa
Separable Equations
We have found a strategy for solving rst-order linear equations, and would now like to consider a
particular class of rst-order nonlinear equations. We will again make use of exact derivatives.
Definition A dierential equation that can