MATH 242 Exam 2
Name:
1
(1) Suppose that the function y1 is a solution to the initial value problem
dy
= (y + 1)(y 1),
dx
(a) Is y1 increasing or decreasing on its domain?
(b) What is lim y1 (x)?
y(0)
DIFFERENTIAL EQUATIONS HOMEWORK 3
DUE 2 OCTOBER
1 Suppose that at time t = 0, ten thousand people in a city with population M = one hundred
thousand have heard a certain rumor. After 1 week, the numbe
In 1-3, nd the Laplace transform of the given function.
1) f (t) = 3 sin(2t) + 4 cos(3t)
Solution
F (s) = 3
s
2
+4 2
.
+4
s +9
s2
2) f (t) = te2t
Solution
2t st
F (s) =
te e
te(s2)t dt.
dt =
0
0
Let u
DIFFERENTIAL EQUATIONS 6
DUE 16 NOVEMBER
Write the solution set for each of the following dierential equations.
1) y y = 0. Note that x2 1 = (x + 1)(x 1).
Solution The solutions of the polynomial are
HOMEWORK ASSIGNMENT 4
In 1 3, perform three steps of Eulers method with step size h = .1
1) y = x + y, y(0) = 2.
Solution We have x0 = 0, y0 = 2, and f (x, y) = x + y. Then
y1 = f (x0 , y0 )(.1) + y0
HOMEWORK ASSIGNMENT 2 SOLUTIONS
Find the general solutions to the following dierential equations:
1: x3 + 3y xy = 0
3
Solution Rearranging gives y x y = x2 . This is linear with integrating factor x3
In problems 1-4, nd all solutions of the given dierential equation.
1) y (5) 2y (4) + y (3) = 0.
Solution Write the polynomial
x5 2x4 + x3 = 0.
This factors as x3 (x 1)2 . The roots are x = 0 with mul
MATH 242 Exam 2
Name:
1
(1) Suppose that the function y1 is a solution to the initial value problem
dy
= (y + 1)(y 1),
dx
(a) Is y1 increasing or decreasing on its domain?
(b) What is lim y1 (x)?
y(0)
DIFFERENTIAL EQUATION EXAM 3 PREPARATION
Solve the following homogeneous linear dierential equations:
1) y 2y + 2y = 0.
2) y 4y + 4y = 0.
3) y 4y + 3y.
4) y y + y y = 0, noting that x3 x2 + x 1 = (x 1
MATH 242 Exam 1
Name:
1
In 1-5, solve the given dierential equation.
1) (x+y)y = xy
2)
dy
= yex
dx
2
3) xy 3y = x3
4)
x3 +
y
dx+(y 2 +ln x)dy = 0
x
3
5) x
dy
+6y = 3xy 4/3
dx
In problem 6, reduce the
HOMEWORK ASSIGNMENT 1
DUE WEDNESDAY, 9 SEPTEMBER
Find the solutions of the given dierential equations (implicit if necessary).
1: y + 2xy = 0
Solution This is separable. We may rewrite as y = 2xy and