Power Series Solutions: Solve ODEs Step by Step Easily

A power series solution of a differential equation is an assumed solution of the form y(x) = a0 + a1x + a2x2 + …, written compactly as y = Σ anxn, whose coefficients are found by substituting the series into the equation and matching the coefficients of like powers of x.

This matters because many equations from real physics have no solution in elementary functions. Airy's equation y″ − xy = 0, which describes light near a caustic and quantum particles at a turning point, cannot be solved with sines, exponentials, or polynomials — but a power series builds its solution term by term and evaluates it to any desired accuracy.

Key Concepts

• Ordinary point: x0 is ordinary for y″ + P(x)y′ + Q(x)y = 0 if P and Q are analytic (have their own convergent power series) at x0. The method works at ordinary points.
• Singular point: a point where P or Q blows up. Ordinary power series fail there; the Frobenius method is needed instead.
• Recurrence relation: the equation linking the coefficients, typically expressing an+2 in terms of earlier coefficients.

Formulas

• y = Σn=0∞ anxn, y′ = Σ n·anxn−1, y″ = Σ n(n−1)anxn−2.
• Index shifting: Σn=2∞ n(n−1)anxn−2 = Σn=0∞ (n+2)(n+1)an+2xn, so every series is rewritten in powers xn before matching coefficients.
• Initial data fix the first coefficients: a0 = y(0), a1 = y′(0); the recurrence then determines all the rest.
• Assumptions: P(x) and Q(x) analytic at x0. Limit: the series represents the solution only inside its radius of convergence.

Worked Example

Solve y″ + y = 0 with y(0) = 1, y′(0) = 0 by series. Substitute: Σ n(n−1)anxn−2 + Σ anxn = 0. Shift the first sum: Σ (n+2)(n+1)an+2xn + Σ anxn = 0. Matching coefficients of xn gives the recurrence an+2 = −an/((n+2)(n+1)). From the initial conditions a0 = 1 and a1 = 0. Then a2 = −1/(2·1) = −1/2, a3 = −a1/6 = 0, a4 = −a2/(4·3) = (1/2)/12 = 1/24. Hence y = 1 − x2/2 + x4/24 − …, which is the Taylor series of cos x — exactly the solution the characteristic-equation method gives.

Common Misconceptions

• It works for any ODE at any point. False: at a singular point the plain series ansatz collapses and you need the Frobenius method with its extra xr factor.
• The series equals the solution everywhere. Only within the radius of convergence. For y′ = y the series for ex converges everywhere, but a series about x = 0 for an equation singular at x = 1 stops being valid past x = 1.
• The equation determines a0 and a1. No — they come from the initial conditions. The recurrence only produces a2 onwards.

Real-World Applications

• Airy functions in optics (diffraction near caustics) and quantum mechanics are defined and computed by their power series.
• Bessel functions, used for vibrations of circular drums and heat flow in cylinders, were historically tabulated from series expansions.

A power series solution assumes y = Σ anxn at an ordinary point, substitutes it into the ODE, and reads off a recurrence relation for the coefficients, with a0 and a1 set by the initial conditions. It solves variable-coefficient equations that have no elementary closed form and doubles as a numerical method.