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
We found out that
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)=y0 and y'(0)=y'0.
Let r>0. If both p(t) and q(t) have Taylor series, which converge on the interval (-r,r), then the differential equation has a unique power series solution y(t), which also converges on the interval (-r,r).
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 p(t) and q(t).
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 y1(t) and y2(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
Legendre's Equation of order
has the form
Be careful! We have to rewrite this equation to be able to apply Fuchs's Theorem. Let's divide by 1-t2:
What is the radius of convergence of the power series representations of
The geometric series
Bessel's Equation of order
has the form
Once again we have to be careful! Let's divide by t2:
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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