Example. Solution of the Bessel
Equation with v = 1
3
August 18, 2006
Solve the Bessel equation
x2 y 00 + xy 0 + x2
1
9
y
x0
0.0.1
=
0
(1)
= 0
Step 1.
x = 0 is the singular point.
1
1
Here P (x) = x ; Q(x) = 1 9x2
2
p (x) = xP (x) = 1; q (x) = x Q (x) = x

DIFFERENTIAL EQUATIONS
I. Introduction
What is a dierential equation? An equation for one or more unknown functions involving
derivatives of these unknowns, e.g.,
1.
2.
3.
4.
y + xy + 3y 2 = ex ;
u + u = cos(t);
uxx + uyy = 0;
x = y, y = x.
If the unknown

Chapter 1
Introduction and rst-order
equations
In this introductory chapter we dene ordinary dierential equations, give
examples showing how they are used and show how to nd solutions of some
dierential equations of the rst order.
1.1
What is an ordinary

113
Spotlight on the Extended Method of Frobenius
See Sections 11.1
11.2 for the model of
and
an aging spring.
Reference: Section 11.4 and S POTLIGHT ON B ESSEL F UNCTIONS.
Bessel functions of the rst kind were constructed early in the nineteenth century

Copyright
1999 University of Cambridge. Not to be quoted or reproduced without permission.
Copyright
1999 University of Cambridge. Not to be quoted or reproduced without permission.
Copyright
1999 University of Cambridge. Not to be quoted or reproduced wi

Copyright
1999 University of Cambridge. Not to be quoted or reproduced without permission.
Copyright
1999 University of Cambridge. Not to be quoted or reproduced without permission.
Copyright
1999 University of Cambridge. Not to be quoted or reproduced wi

Copyright
1999 University of Cambridge. Not to be quoted or reproduced without permission.
Copyright
1999 University of Cambridge. Not to be quoted or reproduced without permission.
Copyright
1999 University of Cambridge. Not to be quoted or reproduced wi

Common Derivatives and Integrals
Common Derivatives and Integrals
Derivatives
Integrals
Basic Properties/Formulas/Rules
d
( cf ( x ) ) = cf ( x ) , c is any constant. ( f ( x ) g ( x ) ) = f ( x ) g ( x )
dx
d n
d
( c ) = 0 , c is any constant.
( x ) = nx