Power Series Solutions Near Ordinary Points

The constant-coefficient methods reduce $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$ to algebra through the substitution $y=e^{rt}$. Most equations of mathematical physics have variable coefficients, and no exponential guess works. Two recurring examples are

which arise from heat flow and wave motion in cylinders and from potential theory in spheres.¹ Their coefficients are polynomials in $x$, and no exponential guess works. The replacement strategy is older than either equation: assume the solution is a power series

substitute it into the equation, and solve for the coefficients $a_n$ one at a time. The method is guaranteed to work near an ordinary point of the equation, and the interval on which the resulting series converges can be read off in advance.

Power series preliminaries

A power series about $x_0$ is an expression ${\sum }_{n=0}^{\infty }{a}_n(x-x_0{)}^n$. The full theory belongs to calculus; the working facts are these.²

• Convergence set. Every power series has a radius of convergence $\rho \in [0,\infty ]$: the series converges absolutely for $|x-x_0|<\rho$ and diverges for $|x-x_0|>\rho$. The ratio test computes $\rho$ when the limit ${\lim }_{n\to \infty }|a_{n+1}/a_n|$ exists.
• Termwise calculus. Inside $|x-x_0|<\rho$ the series may be differentiated and integrated term by term; the derived series have the same radius of convergence.
• Algebra. Two series about the same point may be added, multiplied, and (where the denominator is nonzero) divided, with the results valid on the common interval of convergence.
• Identity theorem. If $\sum a_n(x-x_0{)}^n=\sum b_n(x-x_0{)}^n$ for all $x$ in some open interval, then $a_n=b_n$ for every $n$. In particular, a power series that sums to zero on an interval has every coefficient equal to zero. The whole method rests on this fact.

Shifting the index of summation

The method requires rewriting sums so that every series displays the same generic power of $x$. The rule: replacing $n$ by $n+k$ inside the sum, while starting the count $k$ lower, leaves the series unchanged. For example,

as writing out the first few terms of each side confirms.³ Combined with the identity theorem, index shifting already solves equations. Suppose a function satisfies $y^{\prime }=y$ with $y=\sum a_n{x}^n$. Then

so $(n+1)a_{n+1}=a_n$ for every $n$, giving $a_n=a_0/n!$ and $y=a_0\sum x^n/n!=a_0{e}^x$. The same procedure works for any first-order linear equation at a point where $Q/P$ is analytic.⁴

Ordinary and singular points

Fix the homogeneous second-order equation

where $P$, $Q$, $R$ are polynomials with no factor common to all three (if there is one, divide it out first).

Near an ordinary point the equation divides through by $P$ to give $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=0$ with continuous coefficients, so the existence and uniqueness theorem supplies a unique solution for any initial data $y(x_0)=y_0$, $y^{\prime }(x_0)=y_0^{\prime }$. This solution is analytic, not merely continuous, and can be computed term by term.

Which points are singular is decided in the complex plane, not just on the real line, and this matters for convergence. The equation $(1+x^2)y^{\prime \prime }+2x{y}^{\prime }+4x^{2}y=0$ has no real singular points at all, yet $P(x)=1+x^2$ vanishes at $x=\pm i$, and those complex zeros limit every real series solution centered at the origin to radius $1$.^boyce-part2

The method at an ordinary point

Let $x_0$ be an ordinary point. Assume a solution

convergent on some interval $|x-x_0|<\rho$, differentiate termwise, substitute into the equation, expand the coefficients $P,Q,R$ in powers of $x-x_0$ where needed, shift indices until one generic power remains, and set each coefficient to zero. The result is a recurrence relation expressing later coefficients in terms of earlier ones, while $a_0$ and $a_1$ stay free and fix the initial data $y(x_0)$ and $y^{\prime }(x_0)$.

Take the equation $y^{\prime \prime }+y=0$, where every point is ordinary ($P=1$) and the answer is known in advance, as a first test of the method.

Both series pass the ratio test for every $x$, which justifies the termwise operations. The two bracketed series satisfy $y_1(0)=1,y_1^{\prime }(0)=0$ and $y_2(0)=0,y_2^{\prime }(0)=1$, so their Wronskian at $0$ is $1$ and they form a fundamental set. Nothing in the computation used prior knowledge of sine and cosine; had these functions been unknown, the series would define them, and their standard properties ($S^{\prime }=C$, $C^{\prime }=-S$, parity) all follow from the series.⁶ Several classical special functions have no more elementary definition than this.

Airy's equation $y^{\prime \prime }-xy=0$ has non-constant coefficients and solutions that are not elementary functions, so the series is the only way in.

The solutions oscillate with growing frequency and shrinking amplitude for $x<0$ and grow monotonically for $x>0$. The behavior is visible from the equation itself: $y^{\prime \prime }=xy$ makes solutions curve away from the axis where $x>0$ (exponential character) and toward it where $x<0$ (oscillatory character). Airy functions are not expressible in elementary terms, but as solutions of an initial-value problem they are concrete computational objects.

Accuracy of truncated series

The series is exact; any finite partial sum is a local approximation, accurate near $x_0$ and worthless far away. The polynomial approximations to $\cos x$ show the pattern: each added pair of terms extends the interval of useful accuracy, but every polynomial eventually leaves the bounded oscillation behind.

A practical error estimate needs no theory: plot two consecutive partial sums and trust the approximation only where their graphs have not yet separated.⁷

Radius of convergence

Fuchs settled the convergence question in 1866.⁹

For polynomial coefficients the lower bound becomes geometric. If $Q/P$ is in lowest terms, its Taylor series about $x_0$ converges out to the nearest zero of $P$ in the complex plane. So:

The bound is computed by inspection, before any coefficients are found. Worked instances:^boyce-part2

Two remarks:

• It is only a lower bound. For Legendre's equation with $\alpha$ a nonnegative integer, one of the two series terminates: the solution is a polynomial and converges for all $x$, far beyond the guaranteed $|x|<1$. These polynomial solutions are the Legendre polynomials of the special-functions lesson.
• Analytic, not just polynomial, coefficients work. For $y^{\prime \prime }+(\sin x)y^{\prime }+(1+x^2)y=0$, both coefficients are entire (their Taylor series converge everywhere), so series solutions about $0$ converge for all $x$ even though the coefficients are not polynomials.

The equation has no real singular points, so the basic existence theorem gives a unique solution on all of $(-\infty ,\infty )$; the series theorem only guarantees a power-series representation of that solution on the disk above. The solution exists globally, but its Taylor series about a point need not converge globally. Existence of the solution and convergence of one particular representation of it are different questions.

Failure at singular points

At a singular point $P(x_0)=0$, the coefficient functions $p,q$ blow up, Fuchs' theorem gives no guarantee, and the solutions themselves typically fail to be analytic: near a singular point they may grow without bound, oscillate infinitely fast, or involve fractional powers and logarithms that no Taylor series can represent. Singular points cannot simply be avoided. In applications they are usually where the geometry concentrates (the center of the membrane, the ends of the interval), and the physically relevant question is how solutions behave as $x\to x_0$. The singular points mild enough to handle are the regular singular points, and the power series generalizes there to the Frobenius form $x^r\sum a_n{x}^n$, treated in the Frobenius method.