MA205 Lab 6 Report - Series Solutions of Dierential Equations
Name:
Student Number:
Lab
Winter 2015
1. [8 marks] Consider the Van der Pol Equation:
y +
1
y 2 1 y + y = 0 with initial values y(0) = 1, y (0) = 0
10
The Van der Pol equation has applications
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 11 LESSON 21 & 22
8.3 Power Series
Solutions to Linear
Differential Equations
Setup and Standard Form
We will be looking at differential equation
2 ()
2
2
+ 1 ()
+ 0 () = 0,
And write it in standard form
(1
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 10 LESSON 20
8.2 Power Series and
Analytic Functions
Power Series
A power series about the point 0 is an expression of the form
(1)
=0
0 ,
where x is a variable and the a 's are constants.
We say that (1) co
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 6 LESSON 11
6.3 Undetermined
Coefficients and the
Annihilator Method
Setup
Let an , 1 , , 1 , 0 be real numbers and 0
an
+
1
1 1
+ + 1
+ 0 = ()
Annihilator Method
Definition: A linear differential operato
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 8 LESSON 15
7.4 Inverse Laplace
Transform
Inverse Laplace Transform
Linearity of Inverse Transform Theorem
Example
4 + 4 = e3x
Method of Partial Fractions
We turn
()
into factors that we can inverse.
1. No
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 9 LESSON 18
11.2 Series
Infinite Series
Definition: Given a series
=1
= 1 + 2 + 3 + , let sn denote its nth partial sum:
=
= 1 + 2 + +
=1
If the sequence cfw_ is convergent and lim = exists as a real num
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 7 LESSON 13
6.4 Method of Variation
of Parameters
Set Up
We look at
+ 1
1
1
+ + 1
+ 0 = ()
where the functions 1 (), , 1 (), 0 () and () are continuous on and let
cfw_1 , , () be a fundamental solution
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 8 LESSON 16
7.5 Solving Initial Value
Problems
Method of Laplace Transforms
To Solve an initial value problem:
a) Take the Laplace transform of both sides of the equation.
b) Use the properties of the Laplace
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 9 LESSON 17
7.6 Transforms of
Discontinuous and
Periodic Functions
Basic Functions
Unit Step Function
Definition: The unit step function is defined by
0, < 0
1, 0 <
Rectangular Window Function
Definition: T
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 7 LESSON 14
7.2 Definition of the
Laplace Transform
Laplace Transform
Linearity of the Transform Theorem
Existence of Transform
You can not take the Laplace Transform of every function as some grow to fast. W
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 10 LESSON 19
8.1 Introduction: the
Taylor Polynomial
Approximation
Taylor Polynomial of Degree n
The formula for the Taylor polynomial of degree n centered at 0 , approximating a function f(x)
possessing n de
Notes on Method of Frobenius
Robert Rundle
April 7, 2015
The problem with the example done in class was that when I was taking
the derivatives of the power series I always removed a value from the index. So
y(x) = n=0 an xn+r was then y (x) = n=1 (n + r)a
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 12 LESSON 23 AND 24
8.6 Method of
Frobenius
Setup
Regular Singular Point
Example
1.
2 + 3 6 = 0
2.
sin
3.
cos x
x
4.
2 1 2 = 0
Indicial Equation
Method of Frobenius
Method of Frobenius Continued
Frobeniuss
MA205 Lab Report 4 Prep - Linear Dierential Equations - Part 2
Homogeneous Dierential Equations with Constant Coecients:
dn y
dn1 y
dy
+ an1 n1 + . + a1
+ a0 y = 0
where ai are constants for all 0 i n
n
dx
dx
dx
nd
General Solution (very similar to method
MA205 Lab Report 6 Prep
1. Power Series:
In general, the Power series about the point x = x0 is given by:
an (x x0 )n = a0 + a1 (x x0 ) + a2 (x x0 )2 + .
n=0
denes a function of x (an innite polynomial), the domain of which is all x for which the series
MA205 Lab 5 Report - Laplace Transforms
Name:
Student Number:
Lab
Winter 2015
1. [8 marks] Suppose f (t) is the hyperbolic cosine function given by:
f (t) = cosh(t) =
et + et
2
(a) Use the Denition 1 (on page 353 of the text of in the lab prep) to show th
MA205 Lab Report 1 Prep
1. Linear Ordinary Dierential Equations:
dn1 y
dy
dn y
+ an1 (x) n1 + . + a1 (x)
+ a0 (x)y = g(x)
dxn
dx
dx
Has Order n with independent variable x and dependent variable y
General Form: an (x)
Example:
dh
, in a draining tank is
MA205 Lab 3 Report - Linear Dierential Equations - Part 1
Name:
Student Number:
Lab
Winter 2015
1. [12 marks] Find the general solution to each of the following linear homogeneous dierential equations with
constant coecients.
(a) 2
d2 y
18y = 0
dx2
Solvi
MA205 Lab Report 3 Prep
Homogeneous Dierential Equations with Constant Coe:
d2 y
dy
For DEs of the form: a 2 + b
+ cy = 0
where a, b,and c are constants
dx
dx
To nd the general solution y(x):
Determine the roots of the auxiliary equation ar2 + br + c = 0
MA205 Lab Report 2 Prep
First Order Dierential Equations:
1. Separable Equations:
Of the form:
dy
= g(x)h(y)
dx
Solution: Re-arrange the D.E. into the form
dy
=
h(y)
dy
= g(x)dx
h(y)
g(x)dx
Integration will result in an implicit solution in terms of y and
MA205 Lab 2 Report - First Order Dierential Equations
Name:
Student Number:
Lab
Winter 2015
1. [2 marks] Indicate (by circling) whether each of the following dierential equations is separable, linear, exact,
or a combination of the three. Extra space is p
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 6 LESSON 12
4.6 Variation of
Parameters
Setup
We begin our study of the linear second-order constant-coefficient differential equation
2
2
+
+ = , 0
with 0.
We have the auxiliary equation
2 + + = 0
and we
MA 205 Section B
DIFFERENTIAL EQUATIONS I WEEK 5 LESSON 10
4.4 Nonhomogeneous
Equations: The Method of
Undetermined Coefficients
Setup
We begin our study of the linear second-order constant-coefficient differential equation
2
2
+
+ = , 0
with 0.
Exampl