Poulin, F.J.
AMATH 351
1: (18 points)
a) The Wronskian is dened to be
W (y1 , y2 ) = y1 y2 y2 y1
b) We dierentiate and substitute in the equation to nd,
W = y1 y2 y2 y1 ,
= P W.
We can solve this equation to get,
x
W = W0 exp
P (s) ds
0
W = 0 if and only
AMATH 351: Assignment 6:
1.
a) Integrate the equation from x = 0 to x and get,
x
y(s) ds.
y(x) = y(0) + A
0
b) We dene the iterative formula,
x
yn+1 (x) = y0 + A
yn (s) ds.
0
and then apply Picards method to get
y0 = y0 ,
x
y0 (s) ds = (I + xA) y0 ,
y1 (x
AMATH 351: Solutions for Assignment 2:
1.
a) First try sin x,
sin2 x 2 cos2 x sin2 x = 2 = 0.
This is not a solution so next try cos x,
sin x cos x + 2 cos x sin x sin x cos x = 0.
Therefore, y1 (x) = cos x is one solution.
b) Then we can use reduction
AM 351: Assignment 2:
Due October 2, 2015
1. The following equation arises in the mathematical modelling of reverse osmosis,
(sin x)
d2 y
dy
2(cos x) (sin x)y = 0,
dx2
dx
on
0 < t < .
Find a general solution to this homogeneous equation.
[Hint: The form
AM 351: Assignment 5
Due November 16th, 2012
1. Nullclines
a) Consider the system
d
s1 (t) = V k3 s1 (t) + k5 s2 (t)
dt
d
k1
k5 s2 (t) k4 s2 (t)
s2 (t) =
dt
1 + sN (t)
1
which describes the concentrations of two species in a biochemical network. Take par
AM 351: Assignment 2:
Due 11:30am, Oct 5th, 2012
1. Heated spring Recall that the equation
my (t) + by (t) + ky (t) = 0
describes the position y of a mass attached to a wall by both a spring, with spring constant
k , and a damper, with damping constant b.
AM 351: Assignment 4 Revised
Due November 2, 2012
1. For the following nonlinear systems, identify all steady states of the system and classify their
dy
stability type. Then, use the fact that the ratio dx = dy / dx is separable to arrive at explicit
dt d
AM 351: Assignment 3: Solutions
Due October 24th, 2012
1. Find the fundamental matrix (t, 0) associated with the linear system
3 2
2 3
a) A =
,
1 4
1 1
b) A =
dx
dt
= Ax(t) where,
.
SOLUTION.
a) The eigenvalues of this matrix are 1 = 5, = 1 with correspon
AM 351: Assignment 4
Due October 31th, 2012
1. For the following nonlinear systems, identify all steady states of the system and classify their
dy
stability type. Then, use the fact that the ratio dx = dy / dx is separable to arrive at explicit
dt dt
desc
AM 351. Spring 2015. Assignment 1.
Topics: Linear independence, reduction-of-order, and variation-of-parameters
Due May 22nd, 2015
1. (a) In the derivation of the variation of parameters formula, we made the simplifying as
sumption that 1 1 + 2 2 = 0, whe
AM 351: Assignment 4
Due June 19th, 2014
1. Harmonic oscillator + 2 = 0.
(a) Write this 2 -order equation as a system of 1 -order equations.
(b) Use the fact that = sin + cos solves the 2 -order equation to write out the
fundamental matrix () correspondin
AM 351: Assignment 7
Due December 7, 2012
1. Suppose that the transfer function of a linear time-invarant system is,
G(s) =
(s2
s
.
+ 1)(s + 2)
nd the response to the system in the time domain to,
a) a unit impulse at t = 0.
b) a unit step input at time t
AM 351. Fall 2012. Assignment 1.
Due Sept. 26th, 2012
1. IVPs vs. BVP: Consider the simple harmonic oscillator
y + a2 y = 0,
(1)
with a natural frequency of a > 0.
(a) What is the general solution of this DE? Denote it as yh (x).
(b) Now consider the IVP
AM 351: Fall 2012. Assignment 1: Solutions
1. IVPs vs. BVP: Consider the simple harmonic oscillator
y + a2 y = 0,
(1)
with a natural frequency of a > 0.
(a) What is the general solution of this DE? Denote it as yh (x).
(b) Now consider the IVP that consis
AM 351: Assignment 2: Solutions
1. Heated spring Recall the that equation
my (t) + by (t) + ky(t) = 0
describes the position y of a mass attached to a wall by both a spring with spring constant
k and a damper with damping constant b. When the spring is he
AMATH 351 - Ordinary Dierential Equations 2
Midterm Exam II - Fall 2012
Instructor: Brian Ingalls
Friday, Nov. 9th, 2012, 10:30-11:20
Materials: Two double-sided sheets of notes. Calculator.
Do your best to answer the questions clearly and carefully. Use
AMATH 351 - Ordinary Dierential Equations 2
Midterm Exam I - Fall 2012 - SOLUTIONS
Ingalls, B.
Date: Friday, Oct. 12th, 2012
Do your best to answer the questions clearly and carefully. Use any appropriate formulas as
needed, but be sure to indicate where
AM 351: Assignment 5
Due November 16th, 2012
1. Nullclines
a) Consider the system
d
s1 (t) = V k3 s1 (t) + k5 s2 (t)
dt
d
k1
k5 s2 (t) k4 s2 (t)
s2 (t) =
dt
1 + sN (t)
1
which describes the concentrations of two species in a biochemical network. Take par
AM 351: Assignment 3
Due October 24th, 2012
1. Find the fundamental matrix (t, 0) associated with the linear system
3 2
2 3
a) A =
2.
,
1 4
1 1
b) A =
dx
dt
= Ax(t) where,
.
a) Find the fundamental matrix (t, t0 ) of the DE,
x (t) = A(t)x(t),
2t 0
1 2t
wi
AM 351: Assignment 6
Due November 23rd, 2012
1. Verify that if F (s) = Lcfw_f , then
Lcfw_tf (t) =
dF
(s).
ds
2. Use the Laplace transform to solve the following initial value problems.
a) x (t) + 4x(t) = sin(t),
x(0) = 0,
b) x (t) + tx (t) x(t) = 0,
x (
AM 351: Assignment 7
Due December 7, 2012
1. Suppose that the transfer function of a linear time-invarant system is,
G(s) =
s
.
+ 1)(s + 2)
(s2
nd the response to the system in the time domain to,
a) a unit impulse at t = 0.
b) a unit step input at time t
AM 351: Assignment 6
Due November 23rd, 2012
1. Verify that if F (s) = Lcfw_f , then
Lcfw_tf (t) =
dF
(s).
ds
SOLUTION: a) We nd, by interchanging the derivative and the integral,
d
d
F (s) =
ds
ds
=
0
est f (t) dt
0
d st
e f (t) dt
ds
test f (t) dt
=
0
AM 351. Spring 2015. Practice problems
Topics: Linear independence, reduction-of-order, variation-of-parameters, series solutions
1. Determine the largest interval in which the given initial value problem is guaranteed to have
a unique analytic solution.
AM 351. Spring 2015. Assignment 1 SOLUTIONS.
1. (a) Writing the particular solution as = 1 1 + 2 2 , in general the rst-derivative is,
= () + 1 1 + 2 2 ,
where we have written 1 1 + 2 2 = (). The second-derivative is then written,
= () + 1 1 + 2 2 +