Math 3D Differential Equations Homework Answers 6-1
7
7.2
Power Series Methods
Series Solutions and 2nd Order ODEs
1 Try the solution y( x ) = =0 an ( x 1)n . Then
n
n =0
n =0
an n(n 1)(x 1)n2 = an+2 (n + 2)(n + 1)(x 1)n
y =
Now if y + y = 0 we have
(an
Math 3D Differential Equations Homework Questions 5
Section 3.5 + Nonlinear Systems
1,2 Solve the linear system and decide whether the critical point (0, 0) is stable or unstable. Sketch
the direction eld and use it to decide whether (0, 0) is a node, a c
Notes on Diffy Qs
Differential Equations for Engineers
by Ji Lebl
r
December 18, 2013
2
A
Typeset in LTEX.
Copyright c 20082013 Ji Lebl
r
This work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0
United States License. To
Extra Practice Problems for the Final
1.) Laplace transform of t sin(8t).
2.) Laplace transform of t2 e3t .
3.) Laplace transform of sin(t)u(t 5).
4.) Inverse Laplace transform of
1
s2 +4s+9 .
5.) Use Laplace
transforms to solve x00 + 6x0 + 9x = f (t), wh
MATH 3D 10:00am Disc. Quiz 6 March 2nd 2017
Name
ID#
l. (5 points) Find the critical points and the linearizations of the system:
36 = 2(1 1)? + 32/, y' = 2x2 +3y.
2(x~l)2%5y =0 2x2+3yz o
2
y= "%(i/) = 7254" 1
14le Fofm (if?)
Jambian=71[4-X4- 3]
WM)
032
005
.1) LET A: (H I) J gince A i5 (20wa #40119le we automacal/Iy know
'fo u; Eigenvalwe OFA are 1,3 3 and 5. A oli/ZCT calcwla'lion Hf also ghow 771,3.
0 M CO I 0
Nowwmeegenvem, (A 13v. 035>(g 2 Z)/Z):(o)
004- 0
@4c2035o 6:0 mdiuce 25*2L=0=?/p:0
Howev
MATH 3D 10:00am Disc. Quiz 1 January 19'Eh 2017
Name
ID#
l. (5 points) Solve 5-3 : 9 for y(0) = 1.
(029
j. , l
A X45xg 2
:1 ,1
B kgsz-in 2
3, 3:
Hmemjzx; 252%
#344? 15
76)21hf%1'33]*53
3,2):
621 2. (5 points) Find an explicit solution for :c-ly' =
MATH 3D 10:00am Disc. Quiz 4 February 9th 2017
Name
ID#
1. (5 points) Solve y + 73/ + 12y = 2864: for y(0) = 0 and y(0) = 0.
rZ-rZr +12 :0
(V +3)(V+4) :0
r: ~3,4
y; : C(63) 6264 2
. (5
90
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ts)
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1nd
th
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ral sol
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Quiz IV - DEs - December 1, 2015 8 Z 00 am
Name: ID:
>3 Please write the solving process in detail.
1. [10 pts] Let A = (g j).
(a) Find e
(b) Solve r =11: ?(o) = (31).
(a) deAlI) = 331 =(L1Xu1)
Al 3[; gljcwose V7211?
+ = 3 at
A I I; ,4]ch Vzai]
r . I I _
MATH 3D 10:003m Disc. Quiz 7 March 9th 2017
Name
ID#
1. (5 points) a) Using the Heaviside function write down the piecewise function that is 1 for t in
the interval [2, 3] and zero everywhere else.
1mm
2 3
There we off 16me 3 an MW 79V 7th5 proHam.
tie2);
Quiz III - DEs November 17, 2015 8:00 am
Name: ID: b) STaiJIe imeoFe, node ) Sink
3% Please write the solving process mm.
1. [10 pts] Consider the system of linear ODE: , C( / @
xi=x14x2, x2=4x17x2
(a) Write the system in matrix form and find the general
MATH 3D 10:00am Disc. Quiz 5 February 23rd 2017
Name
ID#
1. (5 points) Find the general solution of 50/1 : 6x1 + 3232, at; = $1 + 4x2.
fizyf ) Where P:[ 3]
I 4
deP'lI 12-1mm: (l*3)[1~7)
1,23,12 = 7
M F411? fjf Jmmvm 2. (5 points) Describe the behavior
Math 3D Elementary Dierential Equations Homework Answers 4
3.1 Introduction to Systems of ODEs
2 Can solve x2 = x2 immediately to obtain x2 (t) = c1 eJ . Substituting into the rst equation yields
x1 + x1 = c1 e J + t
This is a linear equation with integra
Math 3D Elementary Differential Equations Homework Answers 3
2.3
Higher order linear ODEs
1 y y + y y = 0 has characteristic equation
0 = 3 2 + 1 = ( 1)(2 + 1)
with solutions = 1, i. The general solution is therefore
y( x ) = c1 e x + c2 cos x + c3 sin x
Math 3D Elementary Differential Equations Homework Answers 2
1.6
Autonomous Equations
x
3
2
(a) The phase portrait of x = x2 is drawn. The only critical
point is x = 0. 0 is unstable: strictly 0 is semi-stable, in that
some solutions starting nearby 0 app
Extra problems with solutions for Notes on Diy Qs.
These are extra problems for my book Notes on Diy Qs. They will become part of the
book at some point, when this set is more or less complete.
Work and justications usually not shown in the solutions. Onl
Math 3D: Fall 2013 Final Exam (v1)
1. Find the general solution to the linear differential equation
(8)
dy
+ 2xy = 2x
dx
2.
(a) Solve the initial value problem
(9)
dy
dx
= (2x 1)(y 1)2
y (0) = 2
(b) What is the solution if the initial condition is changed
Math 3D Differential Equations Homework Answers 6
6.1
The Laplace Transform
5 L 3 + t5 + sin t = 3L cfw_1 + L t5 + L cfw_sin t =
6 L a + bt + ct2 = aL cfw_1 + bL cfw_t + cL t2 =
a
s
+
b
s2
3
s
+
+
7 L cfw_ A cos t + B sin t = AL cfw_cos t + BL cfw_sin t =
Math 3D Elementary Differential Equations Homework Answers 1
0.2
Introduction
4 If x = e4t , then x = 4e4t , x = 16e4t , and x
= 64e4t , whence
x 12x + 48x 64x = (64 12 16 + 48 4 64)e4t = 0
as required.
7 If y = erx , then y = rerx and y = r2 erx , whence
10.19 example
A CNN/Time poll of 1003 American adults, conducted by telephone June 19-20, 2002, was designed
to measure beliefs about apocalyptic predictions. One of the results reported was that 59% of the
sample said that they believe the world wil