MAT 239 (Dierential Equations), Prof. Swift
The Method of Undetermined Coecients
3.5, 4.3 and WeBWorK set 12 linear nonhomogeneous
The general solution to a nonhomogeneous ODE L[y (t)] = g (t) is y (t) = yh (t) + yp (t),
where yh (t) is the general soluti
Eulers Method on a Graphing Calculator
by Jim Swift @ NAU
Eulers method is a way to nd approximate solutions to an Initial Value Problem
(IVP), which is a dierential equation with an initial condition. The slope eld
applet linked to on our web site uses E
MAT 239 (Dierential Equations) Prof. Swift
In-Class worksheet on Solutions by Inspection
dy
Recall that the solution to
= k (y A),
dt
each of these problems by inspection.
1.
dy
= 2(y 1),
dt
2.
dy
y
= ,
dt
2
3.
dy
= 2(y + 1),
dt
4.
du
= 2(u + 3),
dt
5.
dy
Theory of Linear Homogeneous ODEs
by Jim Swift @ NAU
Consider the linear homogeneous ODE
L[y] = y + p(t)y + q(t)y = 0
Suppose that p and q are continuous on an interval I, and y1 and y2 are solutions on
I. Then the following are equivalent. (That means th
MAT 239 (Dierential Equations), Prof. Swift
The Method of Undetermined Coecients
3.5, 4.3 and WeBWorK set 12 linear nonhomogeneous
The general solution to a nonhomogeneous ODE L[y(t)] = g(t) is y(t) = yh (t) + yp (t),
where yh (t) is the general solution
MAT 239 (Dierential Equations)
Theory of Linear Homogeneous Second Order ODEs (3.2,3)
Topics:
General solutions, fundamental solution sets, linearly independent functions, and the
Wronskian.
The topic of discussion is all linear homogeneous second order O
MAT 239, Dierential Equations, Prof. Swift
The Final Exam will be Wednesday, Dec. 10, 2003, from 12:30 to 2:30
This is a copy of a previous semesters nal exam.
Three pages of notes with writing on both sides are allowed. A graphing calculator
is expected.
' d ,
1. Find the explicit. solution to the initial value problem y 2: 3:313; 31(0) :2 1.
aw (in?
(b) Sketch the solution and indicate the interval on which the solution is dened.
(This is also called the imterval of existence, or the interval of validi
MAT 239 (Dierential Equations) Prof. Swift
Extra Credit Opportunity
Extra Credit: Worth 3 class points (Do this on paper and turn in Friday in class):
Use the method of Bernoulli Equations (the last problem in set 4-Linear 1st Order)
to nd the general sol
NAME ms wt Cu {k are a? MATH 239 DIFFERENTIAL EQUATIONS
SECTIONQ, :: FALL 2010 :: FAHY
TEST THREE
Make sure to Show all. necessary work. Correct answers without proper supporting work will not receive full credit. If you are.
unclear regarding my expe
MAT 239 DIFFERENTIAL EQUATIONS
FALL 2012
1
e
EX Solve x'
3 2x t .
2
t
2
1
Start with the homogeneous version x'
3 2x .
1
2 r
det( A rI ) det
3
2 r
0 (2 r )(2 r ) 3
0 r2 4 3
0 r 2 1
r 1
1 0
2 1
1
(1)
0
1
2 1
3
r=1
1 0
2 1
1
3 ( 2)