Math 303 Fall 2007 Final Exam Review Answers
1. These can all be seen by directly substituting the solution into the dierential equation. For instance, for part (b), we have that
= et ,
x
v
A = (A )et .
x
v
Therefore (since et = 0), (t) = et is a solutio
Math 303 Fall 2007 Final Exam Review
The nal exam will be cumulative with a slight emphasis on the material
covered since the second midterm. The best way to review is to work through
the rst two midterms and homework-style problems.
On the nal, you may h
1. y = 3e2x . Thus, y = 6e2x .
y + 2y = 6e2x + 2 3e2x = 0.
2. dv = k (250 v ). (where k > 0 for physical reasons. If this were not so, the
dt
car would be able to accelerate to an indenitely large velocity). Equilibrium
solutions occur when 0 = k (250 v )
Math 303: Fall 2007
Midterm 1: Review Sheet
No calculators, notes, or books will be allowed on the mid-term. The exam
will consist of around 7 questions and last 55 minutes. It is essential that you
clearly and neatly show all work in order to receive ful
1. (10 points) Show that y = x4 ex is a solution to the dierential equation
y=
4
+ 1 y.
x
y = 4x3 ex + x4 ex
4
4
( x + 1)y = x x4 ex + x4 ex = 4x3 ex + x4 ex .
Therefore, we have a y such that y = (4/x + 1)y .
2. (10 points) Find the general solution to
d
Math 303 Fall 2007 Midterm 2 Review Answers
1. (3.1: 20-26, 3.2: 1-12) Show whether the following sets of sets of functions
are linearly dependent or independent:
a. cfw_cos 2x, sin 2x
The Wronskian
cos 2x
(cos 2x)
W = W (cos 2x, sin 2x) =
=
sin 2x
(sin 2
Math 303 Fall 2007 Midterm 2 Review
On the midterm, you may have one 8 1 x 11 sheet of paper (one-side) with
2
formulas, notes, and examples (or favorite recipes or .). There are no calculators allowed, and you may not use any other material, including bo
Math 303: Midterm 2 Answers
November 14, 2007
1. The functions y1 = e2x and y2 = e3x are solutions to the dierential equation
y 5y + 6y = 0.
a. (10 points) Verify that y1 and y2 are linearly independent.
W (y1 , y2 ) =
e2x e3x
2e2x 3e3x
= e5x = 0
b. (10 p