Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2014
MATH 53H, PROBLEM SET #4 SOLUTIONS
4.6. Prove that any two (planar) linear systems with the same eigenvalues i ( real and nonzero) are
conjugate. What happens if the systems have eigenvalues i and i with = ?
Solution: If B and B are 2 2 matrices with eige
Ordinary Differential Equations with Linear Algebra
MATH 53
MATH 53 AUTUMN 2011, HOMEWORK 1 SOLUTIONS
1. Textbook Section 1.1 Problem 6
Solution. Set y 0 equal to 0 to nd that the only equilibrium solution is y D 2. The
slope is negative for y < 2 and is positive for y > 2. The direction eld for
y 0 D y C 2 is:
Di
Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2014
MATH 53H, PROBLEM SET #3 SOLUTIONS
6.12. Compute eA (etA ?), where A is given.
2 1
(b) A =
. We have pA () = ( 2)2 + 1, which has roots = 2 i and corresponding
1 2
eigenvectors z = (i, 1)T . Thus A = P DP 1 where
P =
i
1
i
,
1
2+i
0
,
0
2i
D=
P 1 =
1
2
i
Ordinary Differential Equations with Linear Algebra
MATH 53
MATH 53 AUTUMN 2011, HOMEWORK 2 SOLUTIONS
1. Textbook Section 2.2, Problem 4
Solution. Let Q(t) be the amount of salt in the tank, measured in pounds, and
let V (t) be the volume of water in the tank, in gallons. Then,
dQ
= rate in rate out =
dt
= 3[gal/m
Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2014
Math 53 Assignment 2 Solutions
Spring 2014
2.1#15
Separating the equation:
(1 + 2y) dy
=
2x dx
Integrating:
y + y2
= x2 + C
Given the initial condition y = 0 when x = 2
0 + 02
=
22 + C
Hence C = 4 so in conclusion the solutions satisfy
y 2 + y = x2 4
Solv
Ordinary Differential Equations with Linear Algebra
MATH 53

Winter 2011
Mathematics 53, Winter 2011
Solutions to Homework 2
Section 1.2: The dierential equations are all of the form y (t) +
p(t) y (t) = g(t). We will solve them using the method of integrating factors.
Problem 13. Consider the dierential equation y (t) y (t) =
Ordinary Differential Equations with Linear Algebra
MATH 53

Winter 2011
Mathematics 53, Winter 2011
Solutions to Homework 1
Section 1.1:
Problem 8. We are looking for a dierential equation of the form y =
2
ay + b with the property that the solutions all approach the value y = 3 .
2
b
This means that a must be negative, and t
Ordinary Differential Equations with Linear Algebra
MATH 53

Winter 2011
Mathematics 53, Winter 2011
Solutions to Homework 3
Section 2.1.
Problem 29: Our goal is to solve the dierential equation
y (t) =
ay (t) + b
,
cy (t) + d
where a, b, c, d are given constants. Since this is an autonomous dierential
equation, we may apply t
Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2014
The limit set of a trajectory
Denition. Let F : U Rn be continuously dierentiable, and let x(t) be
a solution of the ODE x (t) = F (x(t) which is dened for all t 0. A point
y Rn is an limit point of the trajectory x(t) if there exists a sequence of
time
Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2014
Math 53H
The Jordan Normal Form
17 Apr 2006
We are concerned with the following problem: Given a linear transformation T : V V , where V is an
ndimensional complex vector space (equivalently, given an n n complex matrix A, where V = Cn ), we
wish to nd a
Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2014
Dynamical classication of planar systems
Proposition. Let A be a real 2 2 matrix. Suppose that A as one positive
and one negative eigenvalue. Then the ow generated by A is conjugate to
1 0
.
the ow generated by the matrix
0 1
Proof. Let us denote the eige
Ordinary Differential Equations with Linear Algebra
MATH 53
MATH 53 AUTUMN 2011, HOMEWORK 3 SOLUTIONS
1. Textbook Section 2.5, Problem 16
Solution. Compute:
@y .y e2xy C x/ D e2xy C y2x e2xy
@x .bx e2xy / D b.e2xy C x2y e2xy /
Equality requires that b D 1.
Find a function P D P .x; y/ such that
(
@x P D y e2xy C x
Ordinary Differential Equations with Linear Algebra
MATH 53
MATH 53 AUTUMN 2011, HOMEWORK 4 SOLUTIONS
1. Textbook Section 3.4, Problem 2
Solution. The eigenvalues of
1
1
A=
4
1
are the solutions of the characteristic equation
det(A I ) = (1 )(1 ) + 4 = 2 + 2 + 5 = 0.
So, the eigenvalues are = 1 2i.
= 1 + 2i:
2i
1
Ordinary Differential Equations with Linear Algebra
MATH 53
MATH 53 AUTUMN 2011, HOMEWORK 5 SOLUTIONS
1. Textbook Section 4.3, Problem 12(a)(b)
Solution.
(a) The characteristic equation is
the solutions are:
D
2
C 2 C 2 D 0. The quadratic formula gives that
2
p
4
8
2
D
1i
The general solution is therefore
c1 e
t
c
Ordinary Differential Equations with Linear Algebra
MATH 53
MATH 53 AUTUMN 2011, HOMEWORK 6 SOLUTIONS
1. Textbook Section 4.5, Problem 6
Solution. To solve the equation
y + 2y + y = 2et
we use the method of undetermined coecients. There should be a particular
solution of the equation of the form
yp (t) = Ats et
To
Ordinary Differential Equations with Linear Algebra
MATH 53
MATH 53 AUTUMN 2011, HOMEWORK 7 SOLUTIONS
1. Textbook Section 5.1, Problem 4
Solution. The graph is
ft
2
1
1
2
t
3
and the function is piecewise continuous.
2. Textbook Section 5.1, Problem 12
Solution. A function f .t / is of exponential order if and onl
Ordinary Differential Equations with Linear Algebra
MATH 53
MATH 53 AUTUMN 2011, HOMEWORK 8 SOLUTIONS
1. Textbook Section 5.4, Problem 10
Solution. To solve the IVP
y + 2y + y = 4et
y (0) = 2, y (0) = 1
apply the Laplace transform to both sides of the equation to get
Lcfw_y + 2Lcfw_y + Lcfw_y = 4Lcfw_et
= s2 Y
Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2014
Inhomogenous linear ODE with periodic nonlinearity
Consider an inhomogeneous linear ODE of the form
x (t) = Ax(t) + f (t),
(1)
where A is an n n matrix, and f (t) is a vectorvalued function. We may
think of f (t) as an external force acting on the system
Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2014
The CayleyHamilton theorem
Theorem 1. Let A be a n n matrix, and let p() = det(I A) be the
characteristic polynomial of A. Then p(A) = 0.
Proof. Step 1: Assume rst that A is diagonalizable. In this case, we
can nd an invertible matrix S and a diagonal ma
Ordinary Differential Equations with Linear Algebra
MATH 53

Winter 2011
Mathematics 53, Winter 2011
Solutions to Homework 4
Section 2.5.
Problem 22: The given dierential equation is M (t, y ) + N (t, y ) y = 0,
where
M (t, y ) = (t + 2) sin(y )
and
N (t, y ) = t cos(y ).
Note that My (t, y ) = (t + 2) cos(y ) and Nt (t, y ) =
Ordinary Differential Equations with Linear Algebra
MATH 53

Winter 2011
Mathematics 53, Winter 2011
Solutions to Homework 8
Section 4.4.
Problem 13: To compute the quasiperiod, we need to nd the roots of the
characteristic equation 2 + + 1 = 0. The solutions of this quadratic
equation are
1
i 4 2 .
=
2
Note that we are taci
Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2015
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Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2015
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Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2013
MATH 53 HOMEWORK 7 SOLUTION
Section 4.3 problems 27, 30, 37, 40, 42.
Section 4.4 problem 7, 11.
Section 4.5 problems 1, 3, 13, 17.
Problem 4.3.27 The characteristic equation is
2 + 2 = 0
( + 2)( 1) = 0
= 1, 2
So the general solution is given by
y(t) = C1
Ordinary Differential Equations with Linear Algebra
MATH 53

Spring 2013
MATH 53 HOMEWORK 1 SOLUTION
Section 1.1 problems 6, 9, 15, 32, 34, 39
Section 1.2 problems 5, 15, 17, 20, 28
Problem 1.1.6 The directional eld for y = y + 2 is given by Figure 1.1.11 in the text. From the plot, one
see that is if y(0) > 2, then y(t) as t