We now consider solving the general second order
linear equation in the neighborhood of a regular
singular point x0. For convenience, will will take x0 =
The point x0 = 0 is a regular singular point
Introduction to Ordinary Differential Equations
Syllabus Spring 2013
Course Home Page
Please check the course home page for course information (homework assignments and exams).
The main purpose of this course is to present methods of
finding solutions, and to discuss properties of solutions of
To provide a framework for this discussion, in this section
Recall the free fall and owl/mice differential equations:
v 9.8 0.2v,
p0.5 p 450
These equations have the general form y' = ay - b
We can use methods of calculus to solve differential
equations of this form.
The Laplace transform is named for the French mathematician
Laplace, who studied this transform in 1782.
The techniques described in this chapter were developed
primarily by Oliver Heaviside (1850-1925), an English
A system of simultaneous first order ordinary differential
equations has the general form
x1 F1 (t , x1 , x2 , xn )
x2 F2 (t , x1 , x2 , xn )
xn Fn (t , x1 , x2 , xn )
where each xk is a function of
In this section we examine a subclass of linear and nonlinear
first order equations. Consider the first order equation
f ( x, y )
We can rewrite this in the form
M ( x, y ) N ( x, y )
For example, let M(x,y) = - f (
Let p, q be continuous functions on an interval I = (, ),
which could be infinite. For any function y that is twice
differentiable on I, define the differential operator L by
L y y p y q y
Note that L[
In Chapter 3, we examined methods of solving second order
linear differential equations with constant coefficients.
We now consider the case where the coefficients are functions
of the independent variable,
Some of the most interesting elementary applications of the
Laplace Transform method occur in the solution of linear
equations with discontinuous or impulsive forcing functions.
In this section, we will assume that all functions consid
Complex Roots of Characteristic Equation
Recall our discussion of the equation
ay by cy 0
where a, b and c are constants.
Assuming an exponential soln leads to characteristic equation:
y (t ) e rt ar 2 br c 0
Quadratic formula (or factoring) yield
Recall the nonhomogeneous equation
y p (t ) y q (t ) y g (t )
where p, q, g are continuous functions on an open interval I.
The associated homogeneous equation is
y p (t ) y q(t ) y 0
In this section we will learn the variation
The general theory of a system of n first order linear equations
x1 p11 (t ) x1 p12 (t ) x2 p1n (t ) xn g1 (t )
x2 p21 (t ) x1 p22 (t ) x2 p2 n (t ) xn g 2 (t )
n pn1 (t ) x1 pn 2 (t ) x2 pnn (t ) x
In some applications, it is necessary to deal with phenomena
of an impulsive nature.
For example, an electrical circuit or mechanical system subject
to a sudden voltage or force g(t) of large magnitude that acts
over a short time in
In this section, we focus on examples of nonhomogeneous
initial value problems in which the forcing function is
ay by cy g (t ), y 0 y0 , y 0 y0
Example 1: Initial Value Problem
Differential equations are equations containing derivatives.
Derivatives describe rates of change.
The following are examples of physical phenomena
involving rates of change:
Motion of fluids
Motion of mechan