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 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 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
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
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 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
3.5
Two dimensional systems and their vector fields
Isolated critical points of linear systems
Definition. A critical point of a system of ODEs is isolated if there are no other critical points nearby.
Consider the following linear system with critical po
Non-linear Systems
Linearization
Definition. Suppose P = ( x0 , y0 ) is an isolated critical point of the system
(
dx
dt = f ( x, y )
dy
dt = g ( x, y )
and that f and g are differentiable at P. The linearization of the system at P is the linear system
(
Math 3D Differential Equations Extra Questions for Midterm Answers
1. The general solution is
y( x ) = c1 e x + c2 e x cos x + c3 e x sin x
Differentiating:
y0 ( x ) = c1 e x + c2 e x (cos x sin x ) + c3 e x (sin x + cos x )
y00 ( x ) = c1 e x 2c2 e x sin
Nonlinear Systems: PredatorPrey Models
Assumptions Two species, one feeding on the other
1. Prey population x (t);
Predator population y(t)
2. If no predators, prey population grows at natural rate: for some constant a > 0,
dx
= ax =) x (t) = x0 e at
dt
3
Population Models
Basic Assumption: population dynamics of a group controlled by two functions of time
Birth Rate b(t, P) = average number of births per group member, per unit time
Death Rate d(t, P) = average number of deaths per group member, per unit t
2. Higher Order ODEs
2.6
2.6. Forced Oscillations and Resonance
Forced Oscillations and Resonance
STUDY OPEN BOOK - DONT MEMORISE FORMULAE!
Damped-driven oscillator: mx00 + cx0 + kx = F(t) = F0 cos wt
Damped spring with mass subject to periodic external f
2. Higher Order ODEs
2.4
2.4. Mechanical Vibrations
Mechanical Vibrations
Mass m attached to spring
x = distance to right of equilibrium
FS = force on mass due to spring
x
x = 0, FS = 0
0
x
x > 0, FS < 0
0
x
x < 0, FS > 0
0
FS
FS
2. Higher Order ODEs
2.4.
Existence and Uniqueness (Picards Theorem)
In each case the theorem does not apply
(
f ( x, y) =
1
1 x
dy
dx
= 11x
y (1) = 1
has no solutions
is not defined (let alone continuous) at ( x, y) = (1, 1)
y
2
1
2
1
1
2
x
1
2
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