Differential equations and boundary value problems
MATH 386

Spring 2007
d r USI r X FQiitG l i p q D E k G r P X D E r S E k X l S r QVI l QTtTD l F&t7Y7itFG l FG f sk S P r E r Ew G q g tsr l "sYir l E l V tRiVhQFWTFG S l FvTVY&FRT&hG l $y ES y S X r IIS y X U G D p n w D U DS E D P D P E q
l E D E U D X r P G D rI S X X D
Differential equations and boundary value problems
MATH 386

Spring 2012
SEC. 9.39.4 OVERVIEW SOLVING ODES WITH FOURIER SERIES
[Read this in conjunction with the examples in class and homework.]
Section 9.3
We solve the boundary value problem
x (t) + ax(t) = f (t),
0<t<L
under either
Dirichlet boundary conditions x(0) = x(L)
Differential equations and boundary value problems
MATH 386

Spring 2012
3.8 (SUPPLEMENT) ORTHOGONALITY OF EIGENFUNCTIONS
We now develop some properties of eigenfunctions, to be used in Chapter 9 for Fourier
Series and Partial Dierential Equations.
1. Denition of Orthogonality
b
We say functions f (x) and g(x) are orthogonal o
Differential equations and boundary value problems
MATH 386

Spring 2012
3.2 (SUMMARY) GENERAL SOLUTIONS OF LINEAR EQUATIONS
[Compare all this material with the second order case, in Section 3.1.]
Consider the nth order linear equation
(1)
y (n) + pn1 (x)y (n1) + + p1 (x)y + p0 (x)y = f (x),
where p0 , . . . , pn1 and f are c
Differential equations and boundary value problems
MATH 386

Spring 2012
FAMOUS SECOND ORDER LINEAR EQUATIONS
In Math 385 we concentrate on constant coecient equations. But when using separation
of variables to solve heat and wave and Schrdinger equations in disks, balls or in cylinders,
o
one often ends up at equations with v
Differential equations and boundary value problems
MATH 386

Spring 2012
Project IV Fourier Series
Robert Jerrard
Goal of the project
To develop understanding of how many terms of a Fourier series are required in order to
wellapproximate the original function, and of the dierences between the Fourier series of
functions with
Differential equations and boundary value problems
MATH 386

Spring 2012
Math 285 Spring 2003 Final Exam Answers
Total points: 200. Do only two of #9, 10, 11, 12. (Cross out the
two you are not doing.) Explain all answers. No notes, books, calculators or
computers.
1. [20 points] Solve
2
y 2xy = exx y 2 ,
1
y(0) = .
2
Solution
Differential equations and boundary value problems
MATH 386

Spring 2012
THE FIRST ORDER LINEAR METHOD
Call a Dierential Equation rst order linear if it can be put in the form
dy
(1)
+ P (x)y = Q(x)
dx
for some functions P and Q.
First Order Linear Solution Method.
Step 0. Put the equation into the standard form (1).
Step 1. F
Differential equations and boundary value problems
MATH 386

Spring 2012
UNDETERMINED COEFFICIENTS SUMMARY SHEET
Goal: to solve Ly = f where L is a constant coecient linear dierential operator and f
has the form
f (x) = Pm (x)erx cos(kx)
or
= Pm (x)erx sin(kx)
or
= Pm (x)erx
where Pm is some polynomial of degree m 0, and r R a
Differential equations and boundary value problems
MATH 386

Spring 2007
NAME:
Math 285 Spring 2003  Test 2
Total points: 100. Do all questions. Explain all answers. No notes, books, calculators or computers. 1. [6 points] For the following differential equation, write down the form of the complementary solution yc , and of t
Differential equations and boundary value problems
MATH 386

Spring 2012
SEC. 9.5 HEAT EQUATION AND SEPARATION OF VARIABLES
You should learn everything on this handout, and should also learn how to come up with
these solutions by using Separation of Variables, as covered in class and in the textbook.
Theorem 1 (Heat equation,