SOLUTION TO WRITTEN HOMEWORK 2
1. Problem 1
The dierential equation is clearly separable. Divide the equation by 1 + y 2 and move
dx to the right we get
ydy
= (1 + x2 )dx
1 + y2
Integrate both sides. The left hand side is
ydy u = y 2 1
1 + y2 = 2
du
1
= l
Homework 8
Solutions.
2
1
, so evaluating at
4
3
(2, 1) does not change it. To nd the eigenvalues of the linear system, we
need to get the zeroes of the characteristic polynomial. That is,
1. The Jacobian of the system is J(x, y) =
0=
2
1
4
3
= (2
)( 3
)+
MATH 216 Homework 9 Solution
April 14, 2014
Problem 1
We see that x = x x3 . So in this case k = 1, and = 1, which makes it into the dynamics of a
compressed hard spring.
Letting x = y gives us the associated linear system to be x = y and y = x x3 . Hence
Math 216
Winter 2014
Written Homework #4
Solutions
1. Solve the initial value problem y 3y 4y = 0, y (0) = 0, y (0) = 5.
The differential equation has characteristic equation r 2 3r 4 = (r 4)(r + 1) = 0 with solutions
r = 4 and r = 1. Theorem 1 of 3.3 of
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Math 216: Assignment 6
Due on March 18, 2014
1
Math 216 : Assignment 6
Problem 1
We begin by making the following identications
x = x1 ,
y = x2 ,
, x = x3 ,
and y = x4 .
The second-order system then becomes
x1 = x3 ,
x2 = x4 ,
x3 = x2 + x2 + t2 ,
1
4
(1)
Review of Complex Numbers
Introduction
This is a short review of the main concepts of complex numbers. Complex numbers are used throughout mathematics and its applications. In particular, when we try to solve dierential equations it is often convenient an