Solutions for homework 5
1
Section 4.4
Harmonic Motion
1. In this exercise
(i) use a computer or calculator to plot the graph of the given function, and
(ii) place the solution in the form y = A cos(t ) and compare the graph of
your answer with the plot f
Solutions for homework 4
1
Section 3.4
A resistor (20 ) and capacitor (0.1 F) are joined in series with an electromotive
force (emf ) E = E (t), as shown in the gure.
If there is no charge on the capacitor at time t = 0, nd the ensuing charge on
the capac
MATH 0290 Differential Equations: Exam 1
October 7, 2013
10:00 to 10:50
Instructions
1. Before you begin, enter your name in the space below.
2. Show all your work on the exam itself. If you need additional space, use the backs of the
pages.
3. You may no
Math 0290
Selected homework solutions
Sept. 22 version, (correction of some signs made to problem 44. The previous
version assumed that t = 0 was noon, not midnight)
pg. 35, #4
y 0 = 1 + y 2 ex
Z
Z
dy
= ex dx
1 + y2
arctan y = ex + c
y = tan (ex + c)
pg.
Math 0290
More solutions
pg. 660, # 7/
uxx + uyy = 0; 0 x 1; 0 y 1
u (0; y) = u (1; y) = u (x; 1) = 0
u (x; 0) = sin 2 x
X 00 + X = 0
X (0) = X (1) = 0
We have solve this quite a few times now. The eigenvalues are
solutions are
X = sin n x
Then,
Y 00
!2Y
Math 0290
More solutions
214, # 2.
(s + 9) L (y) =
L (y) =
1
s+1
1
A
B
=
+
:
(s + 9) (s + 1)
s+9 s+1
1 = A (s + 1) + B (s + 9)
= (A + B) s + (9B + A)
A+B =0
9B + A = 1
1
B = ;A =
8
L (y) =
y=
1
8
1 1
1 1
+
8s+9 8s+1
1 9t 1 t
e + e
8
8
#12
s2 L (y)
9L (y)
Math 0290
More solutions
updated with graphs at 12:30 pm, Dec. 5
pg. 643, # 5. The problem is:
ut = 2uxx ; 0 x
u (0; t) = 0; u ( ; t) = 0
u (0; x) = sin 2x sin 4x
; t>0
We try separation of variables:
u (x; t) = X (x) T (t)
and obtain
X (x) T 0 (t) = 2X 0
Math 0290
Homework solutions since 1st exam
187, # 32
y 00 + 0:5y 0 + 4y = A sin !t
The gain is given by formula 7.15:
1
G (!) = q
! 2 )2
(4
y
1
y
:
+
1 2
!
4
y
1
1
0.8
0.875
0.6
0.75
0.4
0.625
0.2
0.5
0.975
0.95
0.925
0.9
0
1.25
2.5
3.75
5
1.5
1.75
2
2.2
Math 0290, Dierential Equations
Class Notes
1. What is a derivative?
Denition:
f (x)
f (x0) = x!x0
lim
x
0
f (x0)
x0
or
f (x0 + h)
f (x0) = lim
h!0
h
0
f (x0)
We will more often use the notation
x (t) for the function. t is the independent variable and x
Math 0290, Dierential Equations
Class Notes
1. What is a derivative?
Denition:
f 0 (x0 ) = lim
x!x0
f (x)
x
f (x0 )
x0
or
f (x0 + h) f (x0 )
h
We will more often use the notation x (t) for the function. t is the independent variable and x is the dependent
Math 0290, fall, 2006
First hour exam
Solutions
1.(10 pts. each) Find the general solution of the following dierential equations:
(a)
xy 0 + 2y =
cos x
x
cos x
2
y0 + y = 2
x
x
R 2
2
dx
2 log x
e x =e
= elog x = x2
x2 y 0 + 2xy = cos x
0
x2 y = cos x
x2 y
Math 0290
Example of overdamped oscillation
Suppose that a spring-mass system is subject to very strong friction forces. For
example, the mass could be moving in a thick liquid, such as oil or molasses. Suppose
that we stretch the spring and then give it
Math 0290 More solutions 163, # 7. y=
y
p
2e
1 t 2
cos 5t
4
1
0.5
0 0 1.25 2.5 3.75 x -0.5 5
-1
# 24. We .rst must determine the spring constant k: This is like example 1.9 in section 4.1. From Newton' law, F = ma; and from Hooke' law, F = kx; and since a
Math 0290, Fall, 2006
Getting the sign right in solving linear equations
In solving linear equations it is easy to get mixed up on the sign.
example:
x0 2x = 0:
Here is an
We might try this in two ways. We could look for an integrating factor.
what we lea
Solution for homework 3
1
Section 3.1 Modeling Population Growth
10. Suppose a population is growing according to the logistic equation
dP
= rP
dt
1
P
K
.
Prove that the rate at which the population is increasing is at its greatest when
the population is
Solutions for homework 2
2. First Order Equations
2.3. Models of Motion
9. A ball having mass m = 0.1 kg falls from rest under the inuence
of gravity in a medium that provides a resistance that is proportional to its
velocity. For a velocity of 0.2 m/s, t
Solutions for homework 6
1
Section 4.7 Forced Harmonic Motion
3. Plot the given function on an appropriate time interval. Use the technique
of Exercise 2 to superimpose the plot of the envelope of the beats in a dierent
line style and/or color.
cos 10t co
Math 0290 - RUBIN
PHASE PLANES - Feb. 20, 2009
For each of the following systems, produce a phase portrait that includes critical points,
clearly labeled nullclines, the directions of ow in each region, and a representative collection of trajectories (sol
!e r :"itu 2.+
r'
S aofiu) ,1
3 . The d eptlro f t he w ell s atisfies : 4 .9t2,w herer i s
d
the amountof time it takesthe stoneto hit the water,
It a lso s atisfies = 3 40s, w here s : 8 - r i s t he
d
runount of time it takes for the noise of the splas
Solutions for homework 7
1
Section 5.1 The definition of the Laplace
Transform
7. Use Denition 1.1 of the Laplace transform to nd the Laplace transform of
each of the following functions dened for t > 0.
f (t) = te2t .
Using the denition and the integrati
Solutions for homework 8
1
Section 5.4 Using the Laplace Transform to
solve Differential Equations
7. Use the Laplace transform to solve the rst-order initial value problem
y + 8y = e2t sin t,
y (0) = 0.
We compute the Laplace tranform of the LHS using th