Miscellaneous Integrals (Answers)
Instructions: For each of the following, you should be able to perform each of the integrations and
dierentiate the result to get back to the original integrand without any errors.
1.
1
dx = 2 tan1 ( x) + C
(1 + x) x
2.
EA4 homework #3
1
1. The basic method for nding the eigenvalues and eigenvectors of a matrix A discussed in class
is to rst compute the eigenvalues using det.A I / D 0, and then compute the eigenvectors
using .A
I /v D 0.
(a) What is an easier way to get
Honors EA4 homework #2
1
1. Consider the problem
(a)
(b)
(c)
(d)
d 2y
dy
C2
C 2y D 0 ; with y.0/ D 0; y 0 .0/ D 1 :
2
dt
dt
Show that the characteristic equation has complex roots, r D i .
Find the solution to the above problem using the two linearly inde
Honors EA4 homework #1
1
1. [Edwards and Penney Section 1.4 # 58] A water tank has the shape obtained by revolving the
curve y D x 4=3 around the y -axis. A plug at the bottom is removed at 12 noon, when the
depth of the water in the tank is 12 ft. At 1:0
Engineering Analysis 4
Methods for Solving Dierential Equations
I. First-order linear systems (4.14.2, 5.15.2, 5.4)
1. General solution to n n system
dx
= A(t)x
dt
is
x = c1 x 1 + c2 x 2 + . . . + cn x n .
2. Is equation of the form
dx
= Ax
dt
where A is
Engineering Analysis 4
Methods for Solving Dierential Equations, Part 2
I. Second order equations (3.13.3, 3.5)
1. Linear, constant coecient and homogeneous? Try exponential solution erx (or ert if
independent variable is t), get quadratic characteristic
Engineering Analysis 4
Methods for Solving Dierential Equations
I. First order equations (Sections 1.11.5, 2.2)
1. Linear? y + p(x)y = g (x) use integrating factor (x) = exp ( p(x) dx)
2. Nonlinear? If separable (maybe after factoring), put dierent variab
Honors Engineering Analysis 4 Practice Exam
1. Find the solution of the problem
2xy
dy
=
+1
dx
1 + x2
with
y (0) = 1
1. Solution: Solving this problem requires using an integrating factor, which in this case
is
2x
exp
dx = exp [log(1 + x2 )] = 1 + x2 .
2
Honors Engineering Analysis 4 Practice Midterm #3
1. Use eigenvalues and eigenvectors to nd the general solution of the system
dx
= 3x + 4y
dt
dy
=
x 3y
dt
1. Solution: First, we need the eigenvalues of the coecient matrix, which come from
3
4
1
3
= 2 +
Honors Engineering Analysis 4 Practice Exam 2
1. Consider a mass-spring damping system with a 1 kg mass and spring constant of 25 N/m.
(a) Find the subsequent motion if the damping constant is 4 N/(m/sec) and the mass initially has a
displacement of 4 m a
Engineering Analysis 4 Exam Solutions
Winter 2013
1. Consider a mass-spring damping system with a 1 kg mass and spring constant of 10 N/m.
(a) Find the subsequent motion if the damping constant is 2 N/(m/sec) and the mass initially has a displacement of -
Honors Engineering Analysis 4 Exam #1 Solutions
Winter 2013
1. A spherical mothball whose radius was originally 1/4 inch was found to have a radius of 1/8 inch after
one month. Assuming that the volume of the mothball is reduced by evaporation at a rate p