Mathematical Tools for Physics

4—Differential Equations 92

andQare not too bad. Specifically this requires that(x−x 0 )P(x)and(x−x 0 )^2 Q(x)have no singularity at
x 0. For example

y′′+

1

x

y′+

1

x^2

y= 0 and y′′+

1

x^2

y′+xy= 0

have singular points atx= 0, but the first one is a regular singular point and the second one isn’t. The importance
of a regular singular point is that there is a procedure guaranteed to find a solution near a regular singular point
(Frobenius series). For the more general singular point there is no guaranteed procedure (though there are a few
tricks* that sometimes work).
Examples of equations that show up in physics problems are

y′′+y= 0

(1−x^2 )y′′− 2 xy′+`(`+ 1)y= 0 regular singular points at± 1

x^2 y′′+xy′+ (x^2 −n^2 )y= 0 regular singular point at zero

xy′′+ (α+ 1−x)y′+ny= 0 regular singular point at zero

(16)

These are respectively the classical simple harmonic oscillator, Legendre equation, Bessel equation, generalized
Laguerre equation.
A standard procedure to solve these equations is to use series solutions. Essentially, you assume that there
is a solution in the form of an infinite series and you systematically compute the terms of the series. I’ll pick the
Bessel equation from the above examples, as the other three equations are done the same way. The parameter
nin that equation is often an integer, but it can be anything. It’s common for it to be^1 / 2 or^3 / 2 or sometimes
even imaginary, but I don’t have to make any assumptions about it for now.
Assume a solution in the form of a :

y(x) =

∑∞

0

akxk+s

Ifs= 0or a positive integer, this is just the standard Taylor series. It often happens however thatsis a fraction
or negative, but this case is no harder to handle than the Taylor series. For example, what is the series expansion

• The book by Bender and Orszag: “Advanced mathematical methods for scientists and engineers” is a very
  readable source for this and many other topics.