In the last section we looked at one of the easiest examples of a second-order linear homogeneous equation with non-constant coefficients: **Airy's Equation**

which is used in physics to model the defraction of light.

We found out that

and

form a fundamental system of solutions for Airy's Differential Equation.

The natural questions arise, for which values of *t* these series converge, and for which values of *t* these series solve the differential equation.

The first question
could be answered by finding the
radius of convergence of the power series,
but it turns out that there is an elegant Theorem, due to
Lazarus Fuchs (1833-1902), which solves both of these questions simultaneously.

Fuchs's Theorem. Consider the differential equation
y''+p(t) y'+q(t) y=0
with initial conditions of the form y(0)=y_{0} and
y'(0)=y'_{0}.
Let
In other words, the
radius of convergence
of the series solution is at least as big as the minimum of the radii of convergence of |

In particular, if both *p*(*t*) and *q*(*t*) are polynomials, then *y*(*t*) solves the differential equation for all
.

Since in the case of Airy's Equation *p*(*t*)=0 and *q*(*t*)=-*t* are both polynomials, the fundamental set of solutions *y*_{1}(*t*) and *y*_{2}(*t*) converge and solve Airy's Equation for all
.

Let us look at some other examples:

**Hermite's Equation** of order *n* has the form

where

**Legendre's Equation** of order
has the form

where is a real number.

Be careful! We have to rewrite this equation to be able to apply Fuchs's Theorem. Let's divide by 1-*t*^{2}:

Now the coefficient in front of

What is the radius of convergence of the power series representations of

(The center as in all our examples will be

since multiplication by a polynomial (-2

The geometric series

converges when -1<

which will be convergent when -1<

will converge on the interval (-1,1). Consequently, by Fuchs's result, series solutions to Legendre's Equation will converge and solve the equation on the interval (-1,1).

**Bessel's Equation** of order
has the form

where is a non-negative real number.

Once again we have to be careful! Let's divide by *t*^{2}:

Now the coefficient in front of

The function
has a singularity at *t*=0, thus *p*(*t*) fails to have a Taylor series with center *t*=0. Consequently, Fuchs's result does not even guarantee the existence of power series solutions to Bessel's equation.

As it turns out, Bessel's Equation does indeed not always have solutions, which can be written as power series. Nevertheless, there is a method similar to the one presented here to find the solutions to Bessel's Equation. If you are interested in Bessel's Equation, look up the section on "The Method of Frobenius" in a differential equations or advanced engineering mathematics textbook.

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