Step Functions
To deal effectively with functions having jump discontinuities, it is very helpful to introduce a function known as the unit step function or Heaviside function. This function is denoted by uc where c 0, and it is defined by:
!0!if!t < c u

GE 207K
Series solutions near Singular points PS
October 30, 2010
Problem 1 For the following differential equation 2xy + y + xy = 0, (1)
show that x0 = 0 is a regular singular point, and find the series solution corresponding to the largest root at the s

GE 207K
Series Solutions Near a Singular Point
March 10, 2011
This page aims to recap our mini-lecture on `series solution near a Singular point'. We wish expand the solutions of a differential equation that appears as P (x)y + Q(x)y + R(x)y = 0, about a

Series sol. near ordinary points w/ given initial conds. Let's revisit the 2 problems that we have solved in class but let's solve them with some initial conditions. We will solve them two using two different methods. 1. Using the results we obtained in o

Series solutions near Ordinary points PS
March 22, 2011
Problem 1 Find the recurrence relation in the differential equation be near the specified point. Also find the first 3 terms in each of the two solutions (unless the series terminates earlier) (1 + x

GE 207K
Series Solutions Near an Ordinary Point
March 3, 2011
This page aims to recap our mini-lecture on `series solution near an Ordinary point'. We wish expand the solutions of a differential equation that appears as P (x)y + Q(x)y + R(x)y = 0, about a

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TABLE 6.2.1 Elementary Laplace Transforms f t) = -1 cfw_Fs) 1. 1 2. eat 3. t n ; n = positive integer
Chapter 6. The Laplace Transform
Fs) = cfw_ f t) 1 , s>0 s 1 , s>a s-a n! , s>0 n1 s p 1) , s p1 a , s 2 a2 s , s 2 a2 a , s 2 - a2 s , s 2 - a2 s>0

Laplace Transform Step functions PS Problem 1 Find the Laplace transform of g(t) = t2 0 t < 2 6 2<t<
April 9, 2011
(1)
Solution: g(t) in terms of the Heaviside function is: g(t) = t2 - u2 (t)(t2 - 6). Evaluating the two separately, we write L t2 = 2 . s3

Step functions Recall the Heaviside step function (or unit-step function) is defined as uc (t) = 0 x<c 1 x>c
April 9, 2011
(1)
Also recall the "t-shifting" theorem (as opposed to the "s-shifting" theorem): L cfw_uc (t)f (t - c) = e-cs L cfw_f (t) . (2)
In

Laplace Transform PS Problem 1 Solve the following initial value problem y + 2y + y = 4e-t , y(0) = 2, y (0) = -1.
April 9, 2011
(1)
Solution: Take the Laplace transform of both sides of the differential equation: s2 Y - sy(0) - y (0) +2 (sY - y(0) + Y
L

Laplace Transform This page aims to recap our mini-lecture on Laplace transform. The Laplace transform of a function f (t), if it exists, is given by
March 24, 2011
L cfw_f (t) =
0
e-st f (t) dt = F (s).
(1)
Furthermore, the original function in Eq. (1) i

System of Eq. Distinct and Real eigenvalues
April 16, 2011
1
Introduction/Motivation
Note: here x is the dependent variable, and t is the independent variable
Suppose we're given the follow system of differential equations and asked to solve it: x1 (t) =

March 26, 2010 The differential equation of interest is:
1 ln x y + 2 y + y = 0 .
Determine the first three non-zero terms in the series
n=0
an (x - 1)n+r
First verify that x0 = 1 is indeed a singular point (show this):
x1
lim
1 (x - 1) 1 = < 2 ln x 2 =0