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 equa
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 me
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)
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
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
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 m
AM 351: Assignment 5 Solutions
1. Re-writing the inhomogeneous function in tersm of the Heaviside-step function,
() = 1 ( ),
where () = 1 for 0 and 0 otherwise. The Laplace transform of the equation i
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
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 pro
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 e
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 rat
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 spr
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 b
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 appropr
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 concentra
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 ma
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
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)
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
=
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)
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 concentra
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 sec
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)
AM 351: Assignment 3:
Due October 9, 2015
1. Solution to Airys equation,
y + xy = 0,
are called Airy functions and have applications to the theory of diraction.
a) Find the rst three terms of each of