M257/316 SolutionsAssignment 1
UBC M257/316 Resources (c) 2014 by Philip D. Loewen
1. (a) Find the general solution:
y + 6y + 13y = 0.
(b) Find all zeros of the function y, given
4y + 12y + 9y = 0,
(c) Find the general solution:
y(0) = 1, y (0) = 4.
y + 9
Math 257/316 Assignment 1 Solutions
Problem 1. Find the solution to the initial value problem for the ODE
dy
1+y
=
dx
1+x
for each of the initial conditions
(a) y(0) = 1,
(b) y(0) = 1,
(c) y(0) = 2
The ODE is both separable and linear, so we can solve it
Math 257/316 Assignment 7 Solutions
1. The concentration u(x, t) of a reactive chemical diusing in one dimension satises
0 < x < 2, t > 0
ut = uxx u,
u(0, t) = 1, u(2, t) = 1
.
u(x, 0) = 0
where the loss term represents a reaction which consumes the chem
Math 257/316 Assignment 8
Due Mon. Mar. 23 in class
1. Consider the wave equation utt = c2 uxx , < x < , with initial position
x + 1 if 1 < x 0
1 x if 0 < x < 1
u(x, 0) = f (x) =
and with initial velocity ut (x, 0) = 0.
0
otherwise
Sketch the shape of t
SOLUTIONS TO PROBLEM SET 2
1.
Problem 4.
Section 4.1
Let a > Z.
Suppose a is even; then a 0 mod 4 or a 2 mod 4. Since 02
22 4 0 mod 4 we conclude a4 0 mod 4.
0 0 mod 4 and
Suppose a is odd; then a 1 mod 4 or a 3 mod 4. Since 12
32 9 1 mod 4 we conclude a4
SOLUTIONS TO PROBLEM SET 1
1.
Problem 4.
Section 1.3
We see that
1
,
2
1
1 2
1
and is reasonable to conjecture
1 2
1
2
,
3
2 3
1
1 2
Q k k1 1
n
k 1
1
n
n1
1
2 3
3 4
3
,
4
.
We will prove this formula by induction.
Base n
1: It is shown above.
Hypothesis:
Practice 2
1. Find out the fundamental set of solutions to the following equations:
(a) y 00 + 6y 0 + y = 0
(b) (D2 + 2D + 5)(D 2)3 y = 0.
2 5
0
(c) y =
y
1 8
(d) x2 y 00 xy 0 + y = 0
2. Find out the general solution to the following equations:
(a) y 00 4
Trinity University
Digital Commons @ Trinity
Books and Monographs
12-2013
Elementary Differential Equations with Boundary
Value Problems
William F. Trench
Trinity University, wtrench@trinity.edu
Follow this and additional works at: http:/digitalcommons.tr
Final Exam Guide
The cumulative final exam will be on Friday, December 18 from 12:0014:30 in ANGU 098. I will hold extra office hours before the exam, the exact
date will be announced later.
You will be allowed to use calculator. Notes and electronic devi
Math 257/316, Homework 3
1. Consider the differential equation
P (x)y 00 + Q(x)y 0 + R(x)y = 0.
Let x = 0 be a regular singular point.
Then, show that the indicial equation for this differential equation is the indicial equation to the
Cauchy/Euler equati
Math 257/316, Homework 5
Due March 6 IN CLASS
Please bring your homework to the section you are registered to
1. Find the full range Fourier series expansion of the functions;
f (x) = x for < x < ,
g(x) = |x|, for < x <
h(x) = x, for 0 < x < , as a fu
Math 257/316, Homework 4
Due February 27 IN CLASS
Please bring your homework to the section you are registered to
Please solve the following heat equations by first seperating variables and solving the appropriate eigenvalue
problems. When solving the eig
Math 257/316, Assignment 1: due in class September 23rd 2016
1. Compute all eigenvalues and eigenfunctions
of x + x =0, x (a) = 0, x(b) = 0 (assume
a < b). Hint: when > 0, then cos( (t a) and sin( (t a) are also solutions of
the homogeneous equation.
2. C
Math 257/316 Assignment 10 Solutions
1. Find the eigenvalues and eigenfunctions of the following Sturm-Liouville problem on
[0, 1]
X (x) + X(x) = 0, X (0) = 0, X(1) = 0,
and use them to solve the following heat equation with mixed BCs and a source term:
u
LECTURE 1: REVIEW OF ORDINARY DIFFERENTIAL EQUATIONS
MINGFENG ZHAO
July 07, 2015
Separable dierential equations
A separable dierential equation has the form of:
y =
dy
= f (x)g(y).
dx
There are two cases:
Case I: If g(a) = 0 for some constant a, then y(x)
LECTURE 4: SERIES SOLUTIONS NEAR ORDINARY, AND REGULAR SINGULAR POINTS
MINGFENG ZHAO
July 10, 2015
Ordinary and singular points
Denition 1. A function f (x) is analytic at x = a if there exists some r > 0 such that f (x) is innitely dierentiable
in (a r,
LECTURE 5: SERIES SOLUTIONS NEAR REGULAR SINGULAR POINTS
MINGFENG ZHAO
July 14, 2015
Series solutions near a regular singular points
Let x = a be a regular singular point of the following second order linear dierential equation:
(1)
P (x)y + Q(x)y + R(x)y