Applied Math 33
Second Exam, 6 November, 2001.
Points are as indicated. Please identify in a clear manner work that you want considered for
credit. You may use your text book and one (1) calculus book for reference. Calculators that
are capable of solving

AM 33, Fall 2001, Third Practice Exam
1. Let
2 ; 0<t<2
.
t ; t2
f (t) =
Compute the Laplace transform of f . Solve the IVP
y (t) + 2y (t) = f (t),
y (0) = 1.
2. Solve the following integro-dierential equation by taking Laplace transform
t
y (t) = 2y (t) +

AM 33, Fall 2001, Second Practice Exam
1. Find the general solution to the dierential equation
y + y 6y = e2x 2.
2. One solution of
x2 y 2xy + (2 9x2 )y = 0
is y1 (x) = xe3x .
Find a second linearly independent solution y2 .
Calculate the Wronskian W (y

AM 33, Fall 2001, Practice Exam
1. Find all critical points of the dierential equation
dy
= 4y y 3 .
dt
Classify each critical point as stable, unstable or semistable.
Let y (t) be the solution to this dierential equation with y (0) = 1. Without solving
t

dy/dt
2
2
y
Figure 1: graph of f (y ) in the rst question
Fall 2001, AM33 Solution to the practice exam
1. Find all critical points of the diferential equation
dy
= 4y y 3
dt
Classify each critical point as stable, unstable or semistable.
Solution The cri

Fall 2001, AM33 Solution to hw 10
1. Section 6.3, problem 9
t , if t 2
0,
elsewhere
f (t) =
f is nonzero only between and 2 , so we only need to integrate in that region:
2
Lcfw_f =
2
(t )est dt
=
=
es
s2
test dt
2
est dt
e2s
(1 + s)
s2
Where the rst in

Solution to AM 33 HW 8
1. 6.1.2.
2
t
(t 1)1
f (t) =
1
0t1
1<t2
2<t3
Solution: f (t) is continous at [0,1) and (1,3].
2. 6.1.5. Find the Laplace transform of each of the following functions:
(a) t
(b) t2
(c) tn , where n is a positive integer.
Solution:
(

Fall 2001, AM33 Solution to hw7
1. Section 3.4, problem 41 We are solving the ODE
t2 y + 3ty + 1.25y = 0
By problem 38 x = log t turns this DE into a constant coecient DE.
x = log t t = ex
dt
= ex = t
dx
By the chain rule
dy dt
dy
dy
=
=
t
dx
dt dx
dt
We

Solutions to Homework No. 6
SECTION 3.3
Problem 4:
Using Theorem 3.3.1, all we have to do is check whether the Wronskian of f (x) = exp(3x) and g (x) =
exp(3(x 1) is identically zero or not. Since W (f, g ) := f g f g = 3(x 1) exp(3x) exp(3(x 1)
3 exp(3x

Solution to AM 33 HW 5
1. 3.1.15. Solve y + 8y 9y = 0,
y (1) = 1,
y (1) = 0
Solution:
the characteristic equation is:
x2 + 8x 9 = 0
x1,2 = 1, 9
let
y = aex1 + be9(x1)
then
y = aex1 9be9(x1)
plug the initial value in:
a+b=1
a 9b = 0
solve it, get
b=
1
,
10

Fall 2001, AM33 Solution to hw4
1. Problem 3, page 84
Sketch the graph of f (y ) versus y , determine the critical points, and classify each one as
asymptotically stable or unstable.
dy
= y (y 1)(y 2)
dt
Solution
The required graph is:
2.5
x*(x-1)*(x-2)
2

Solutions to Homework No. 3
SECTION 2.4
Problem 10:
Let f (t, y ) = (t2 + y 2 )3/2 . Clearly both f and f /y are continuos on all of the ty plane so that the hypotheses of Theorem 2.4.2 are satised everywhere. Note that this does not mean that we have exi

AM33 Homework assignment # 2
1. (2.2.11)
xdx + yex dy = 0
y (0) = 1
Solution:
xex dx = y dy
therefore
xex dx =
y 2
+c
2
=
y dy
xdex
= xex
ex dx
= xex ex
so
y2
= c + ex xex
2
1
since y (0) = 1, so c = 2 , i.e,
y 2 = 1 + 2ex 2xex
y=
2
2ex 2xex 1
the inter

Fall 2001, AM33 Solution to hw1
1. Find the general solution to the following problems:
(a) Problem 1, p38
y + 3y = t + e2t
Solution
This is a linear equation with constant coecients and solved by nding an integrating
factor (t). (t) has to satisfy:
(t)y

Applied Math 33
First Exam, 9 October, 2001.
Points are as indicated. Please identify in a clear manner work that you want considered for
credit. You may use your text book and one (1) calculus book for reference. Calculators that
are capable of solving t