Ordinary Differential Equations for Scientists and Engineers
MATH 331

Summer 2014
These are reference solutions only. Intermediate step(s) may be skipped!
Show your work for the exam.
1. Find the general solution to each of the following differential equations:
(a) y 0 + xy = x3
Solution: It is a first order linear ODE. So let
R
u = ye
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Fall 2015
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Ordinary Differential Equations for Scientists and Engineers
MATH 331

Fall 2015
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Ordinary Differential Equations for Scientists and Engineers
MATH 331

Fall 2015
Math 331
Page 1 of 7
Homework Assignment 7
Notes: Try to answer all the questions by demonstrating all the steps of
your calculations, and return your answers (including this homework sheet)
back to the instructor as a whole package stapled. Make sure tha
Section 2.8: Modeling: Forced Oscillations.
Resonance
March 711, 2016
1 / 23
Forced Oscillations of motions of a massspring system
The forced oscillations is modeled by the nonhomogeneous ODE
my + cy + ky = r(t),
(1)
where m is the mass of the body, c t
Section 1.3: Separable ODEs. Modeling
January 26, 2016
1 / 19
Topics
Section 1.3: Separable ODEs. Modeling
Separble ODEs. Method of separating variables
Modeling: Mixing Problem and Newtons Law of Cooling
2 / 19
A differential equation along with an initi
Section 2.4: Modeling of Free Oscillations of a
MassSpring System
February 2226, 2016
1 / 20
Summary
Free Oscillations of a MassSpring System:
Undamped MassSpring System
Damped MassSpring System:
Case I: Overdamping
Case II: Critical Damping
Case III: Un
Section 1.5. Linear ODEs. Bernoulli Equation.
Population Dynamics
February 2, 2016
1 / 20
Summary
Linear firstorder ODE: Homogeneous Linear ODE and
Nonhomogeneous Linear ODE
Reduction to Linear Form. Bernoulli Equation. Logistic Equation
Population Dynam
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Fall 2015
Suggested Exercises II
October 23, 2015
Find general solution
1. xy + y = x3
y
y + x = x2 (y x3 )dx + xdy = 0
My = 1, Nx = 1 Exact
4
(y x3 )dx = yx x + h(y) =
4
y = x + h (y) = x
= yx
2.
x4
4
= (r + e ) tan()
= r tan() + e tan()
r r tan() = e tan() Sta
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Summer 2014
1. Find the general solution to each of the following differential equations:
(a) y 0 + xy = x3
(b) y 0 + xy = y
xy
(c) y 0 =
x+y
00
(d) y + 6y 0 + 9y = 0
(e) y 00 + 6y 0 + 9y = e3t
(f) y 00 + y = sin t
2. Solve (a)(c) in problem 1 with initial condition
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Summer 2014
HOMEWORK 05 DUE ON DECEMBER 2ND
ORDINARY DIFFERENTIAL EQUATION, FALL 2016
Instructions : (1) Homework must be written up on paper; (2) Please show all
your work; (3) 5 problems will be randomly chosen and graded on a scale of 0 10;
(4) 10 points are for t
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Summer 2014
HOMEWORK 04 DUE ON NOVEMBER 7TH
ORDINARY DIFFERENTIAL EQUATION, FALL 2016
Instructions : (1) Homework must be written up on paper; (2) Please show all
your work; (3) 5 problems will be randomly chosen and graded on a scale of 0 10;
(4) 10 points are for t
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Summer 2014
Homework 4 Solutions
November 15, 2016
6.2.28
Given
L( f ) =
3s + 4
s4 + k 2 s2
we get:
L( f ) =
Now,
L 1 3
3s + 4
s2 ( s2 + k 2 )
s
k
4
+ 2
2
2
k s + k2
s +k
=
1
s2
= 3 L 1
Therefore, using Theorem 3 we have:
1
s
4
k
=
L 1
3 2
+
s
k s2 + k 2
s + k2
=
3
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Summer 2014
Homework 5 Solutions
December 3, 2016
4.3.2
We are asked to find the real general solution of the system
y10
= 6y1 + 9y2
y20
= y1 + 6y2
thus
"
6
y = Ay =
1
0
#
9
y,
6
" #
y
where y = 1
y2
By substituting y = xet and y0 = xet and dropping the exponential
f
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Summer 2014
HOMEWORK 01 DUE ON OCTOBER 7TH
ORDINARY DIFFERENTIAL EQUATION, FALL 2016
Instructions : (1) Homework must be written up on paper; (2) Please show all
your work; (3) 5 problems will be randomly chosen and graded on a scale of 0 10;
(4) 10 points are for th
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Summer 2014
HOMEWORK 03 DUE ON OCTOBER 26TH
ORDINARY DIFFERENTIAL EQUATION, FALL 2016
Instructions : (1) Homework must be written up on paper; (2) Please show all
your work; (3) 5 problems will be randomly chosen and graded on a scale of 0 10;
(4) 10 points are for t
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Summer 2014
HOMEWORK 01 DUE ON SEPTEMBER 23RD
ORDINARY DIFFERENTIAL EQUATION, FALL 2016
Instructions : (1) Homework must be written up on paper; (2) Please show all
your work; (3) 5 problems will be randomly chosen and graded on a scale of 0 10;
(4) 10 points are for
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Spring 2015
Math331 Midterm Practice Problems. Chapter 12.
Jinguo Lian
May 30, 2013
1.1
Method of Integration
Example 1. Solve the ODE by integration.
(1) y + 2 sin(2x) = 0
2 cos(2x)
Solution: y = 2 sin(2x)dx =
+C is a general solution of the equation.
2
2
(2) y + x
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Spring 2015
Math331 Final Practice Problems. Chapter 4 and 6.
Jinguo Lian
May 30, 2013
4.1
System of ODEs as Models in Engineering Applications
Exercise 1. (1)Find out, without calculation, whether doubling the ow rate in above example has the same eect as halng the
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Spring 2015
Jinguo Lian
Math331 Notes
October 1, 2014
2.2.1
Preliminary: Complex Number
Denition 1. The imaginary unit
i=
1, or i2 = 1
A complex number is defined as
z = a + ib, a, b R
The real part of z is Re(z) = a, the imaginary part of z is Im(z) = b.
Remark 2. (
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Spring 2015
Jinguo Lian
Math331 Notes
September 8, 2014
1.2
1.2.1
Directional Fields and Eulers Method
Preliminary
The tangent line to an implicit function
2 2
Find the tangent line to the circle x + y = 1 at the the point (x0 , y0 ) = (
,
)
2 2
Step 1: Find the slop
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Spring 2015
Jinguo Lian
Math331 Notes
January 24, 2014
1
Introduction
Many problems from engineering, Physics, computer science, biology, chemistry, environmental science, economics etc can be processed (or described) as some mathematical models,
which are ordinary d
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Spring 2015
Math331 Midterm Review. Chapter 12.
Jinguo Lian
October 1, 2013
1.1
Method of Integration
dy
Denition 1. For the rst order ODE y = f (x),
= f (x) dy = f (x)dx
dx
f (x)dx y = f (x)dx. Its general solution is y = f (x)dx
dy =
Example 2. y + xex /2 = 0 wi
Ordinary Differential Equations for Scientists and Engineers
MATH 331

Spring 2015
Jinguo Lian
Math331 Notes
February 5, 2014
1.4
1.4.1
Exact ODEs. Integrating Factors
An Exact dierential Equation
Recall what you learnt in Calculus III, for a function of two variables z = u(x, y), then the
total dierential du = u
dx + u
dy, if we set M