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Lecture Section: - - - - -
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Ma221
168
You may not use a calculator, cell phone, or computer while taking this exam. All work must be
shown to obtain full credit. Credit will not be given for work not reasonably supported. Wh
Lecturer - - - - - - - - - Lecture Section: - - - - -
16S
Ma221
You may not use a calculator, cell phon
while taking this exam. All work must be
shown to obtain full credit. Credit will not be given for work not reasonably supported. When
you finish, be s
Ma 221 - Exam II review
Second Order Differential Equations
Form of general solution
yh c1y1 c2y2
where y 1 and y 2 are linearly independent solutions of the homogeneous equation and
y yh yp
where y p is a [particular] so;lution of the non-homogeneous equ
Ma 221 - Exam I review
First Order Differential Equations
Separable Equations
dy
gxpy
dx
hydy gxdx
hydy gxdx
Linear Equations
dy
pxy qx
dx
Integrating Factor
IF e
e
pxdx
pxdx dy pxy e pxdx qx
dx
d
dx
e
pxdx y e pxdx qx
e
pxdx y
ye
e
pxdx qx dx
p
Ma 221
Workshop 3 (Solutions)
Feb 9, 2016
1. A Bernoulli Equation is a first-order (nonlinear) ODE of the form
dy
+ P (x)y = Q(x)y n .
dx
Show that the following equation can be expressed in the form of a Bernoulli Equation.
Then find a one-parameter fami
Ma 221 - Differential Equations
1. Consider the first-order ODE,
(a) Verify that y =
Homework 1 (Solutions)
Due: Jan 28, 2016
4
dy
= y3
dt
27
is a one-parameter family of solutions (C is any constant).
(t + C)3
(b) Solve the initial value problem (IVP) w
Ma 221 - Exam III review
Laplace Transforms
Definition
ft
0 e st ftdt
Fs fs
Calculate Laplace Transform from definition
Properties
y t sy y0
y t s 2 y sy0 y 0
ft aUt a e as Fs
Note: Ut a in the last line is the unit step function. Ut a
0, t a
1, t a.
Table of Laplace Transforms
Fs cfs f s
ft
s0
1
s
1
n!
tn
s
n1
1
s"a
b
2
s b2
s
2
s b2
n!
s " a n1
b
s " a 2 b 2
s"a
s " a 2 b 2
e at
sin bt
cos bt
e at t n
e at sin bt
e at cos bt
nu1 s0
sa
s0
s0
nu1 sa
sa
sa
cfs " a
e at ft
ft " aUt " a e "as Fs
Note: Ut
Ma 221 - Differential Equations
Homework 2
Due: Sept 15, 2016
Name (Printed):
Recitation:
Collaborators:
I pledge my honor that I have abided by the Stevens Honor System.
Sign:
General Instructions: Write up solutions to the following set of questions and
Ma 221
Homework 10
Name (Printed):
Due: Nov 29, 2016
Recitation:
Pledge and Sign:
General Instructions: Write up solutions to the following set of questions and submit in recitation on
the date indicated. Please staple this cover sheet to your solution pa
Ma 221
Homework 9
Due: Nov 17, 2016
Name (Printed):
Recitation:
Pledge and Sign:
General Instructions: Write up solutions to the following set of questions and submit in recitation on
the date indicated. Please staple this cover sheet to your solution pag
Ma 221
Workshop 6 (Solutions)
Mar 8, 2016
1. The following equation models a mass-spring system with a 2 kilogram mass, a spring
constant of 6 Newtons/meter and a damping force that is 8 times the instantaneous velocity.
The system is driven by an externa
Ma 221 - Differential Equations
Homework 5
Due: Oct 13, 2016
Name (Printed):
Recitation:
Collaborators:
I pledge my honor that I have abided by the Stevens Honor System.
Sign:
General Instructions: Write up solutions to the following set of questions and
Ma 221 - Differential Equations
Homework 6
Due: Oct 20, 2016
Name (Printed):
Recitation:
Collaborators:
I pledge my honor that I have abided by the Stevens Honor System.
Sign:
General Instructions: Write up solutions to the following set of questions and
Ma 221 - Differential Equations
Homework 3
Due: Sept 22, 2016
Name (Printed):
Recitation:
Collaborators:
I pledge my honor that I have abided by the Stevens Honor System.
Sign:
General Instructions: Write up solutions to the following set of questions and
Ma 221 - Differential Equations
Homework 4
Name (Printed):
Due: Oct 6, 2016
Recitation:
Collaborators:
I pledge my honor that I have abided by the Stevens Honor System.
Sign:
General Instructions: Write up solutions to the following set of questions and s
Ma 221
Homework 11
Due: Dec 8, 2016
Name (Printed):
Recitation:
Pledge and Sign:
General Instructions: Write up solutions to the following set of questions and submit in recitation on
the date indicated. Please staple this cover sheet to your solution pag
Ma 221 - Differential Equations
Homework 1
Name (Printed):
Due: Sept 8, 2016
Recitation:
Collaborators:
I pledge my honor that I have abided by the Stevens Honor System.
Sign:
General Instructions: Write up solutions to the following set of questions and
Ma 221 - Differential Equations
Homework 2 (Solutions)
Due: Sept 15, 2016
Graders: 30 pts in total. (Problems # 1, 3 and 4). See each problem for scoring details.
1. Consider the IVP, x2
dy
= y xy,
dx
y(1) = 2.
(a) Find the unique solution and express as
Ma 221 - Differential Equations
Homework 5 (Solutions)
Due: Oct 13, 2016
Graders: 30 pts for the entire assignment. See each problem for particular points.
1. For each of the following differential equations, set up the correct form of the particular solu
Ma 221
Homework 9 (Solutions)
Due: Nov 17, 2016
Graders: 30 pts for the entire assignment. See each problem for particular points.
1. Nonhomogeneous Boundary Value Problem - solution not unique.
Consider the boundary value problem, y 00 + 2 y = g(x) on 0
Ma 221
Homework 10 (Solutions)
Due: Nov 29, 2016
Graders: 30 pts for the entire assignment. See each problem for particular points.
1. Fourier cosine and sine series for functions on [0, L].
Consider the function f (x) = x/L defined on the interval 0 < x
Ma 221 - Differential Equations
Homework 1 (Solutions)
Due: Sept 8, 2016
Graders: 30 pts for the entire assignment. See each problem for particular points.
1. Consider the first-order ODE,
(a) Verify that y =
4
dy
= y3
dt
27
is a one-parameter family of
Ma 221 - Differential Equations
Homework 3 (Solutions)
Due: Sept 22, 2016
Graders: 30 pts in total. (Problems # 1, 3 and 4). See each problem for scoring details.
1. A mathematical model for a falling chain in the absence of resistive forces is given by t
Ma 221
Homework 8 (Solutions)
Due: Nov 10, 2016
Graders: 30 pts for the entire assignment. See each problem for particular points.
1. For the following piecewise functions,
Sketch the graph of g(t).
Express g(t) in the form f0 (t)U(t) + f1 (t t1 )U(t t1
Ma 221 - Differential Equations
Homework 4 (Solutions)
Due: Oct 6, 2016
Graders: 30 pts for the entire assignment. See each problem for particular points.
1. Find a homogeneous linear ODE with constant coefficients whose general solution is,
y = C1 cos 2x
Ma 221 - Differential Equations
Homework 7 (Solutions)
Graders: 30 pts for the entire assignment. See
1. Determine the Laplace Transform for f (t) =
Due: Nov 3, 2016
each problem for particular points.
t
2t
0
0t<1
1t<2
.
2 t < +
Graders: 8 pts;
4 pts for
Ma 221
Homework 11 (Solutions)
Due: Dec 8, 2016
1. Complete solution to the IBVP for the Heat Equation.
Consider the following initial-boundary value problem modeling heat flow in a wire.
u
2u
=2 2
t
x
(PDE)
(BC)
for 0 < x < ,
u(0, t) = 0, u(, t) = 0,
u(x
Nov 19, 2015
Ma 124 Midterm
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