Series Solutions & Taylor Series
When closed-form solutions fail, power series succeed. Learn Taylor and Maclaurin series, radius of convergence, power-series solutions of ODEs, ordinary and singular points, recurrence relations, and the Frobenius method for regular singular points.
Many engineering ODEs (Bessel's for drums and waveguides, Legendre's for spheres, Airy's for optics) have no elementary solutions. Expanding the solution as a power series turns the ODE into an algebraic recurrence for the coefficients.
An analogy
A power series is like building a curve from Lego bricks of increasing detail: 1, x, x², x³ … — add enough and you match the shape as closely as you like.
Taylor polynomials
Truncating the series gives approximations: sin x ≈ x − x³/6 for small x. The error is bounded by the next term (alternating series) or Taylor's remainder.
Power-series ODE solutions
Substitute y = Σ aₙxⁿ, shift indices so all terms have xⁿ, and set coefficients to zero → recurrence. Example y″ + y = 0 gives aₙ₊₂ = −aₙ/((n + 2)(n + 1)) → cos and sin.
Ordinary vs singular points
If p(x), q(x) in y″ + py′ + qy = 0 are analytic at x₀, it's ordinary — plain power series work. If (x − x₀)p and (x − x₀)²q are analytic, it's regular singular — use Frobenius.
Convergence and use
The radius of convergence extends to the nearest singularity (possibly complex). Series solutions define special functions (Bessel, Legendre) tabulated and coded in libraries.
Formulas
- Taylor series:
f(x) = Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n! - Maclaurin examples:
eˣ = Σ xⁿ/n!, sin x = x − x³/3! + …, 1/(1 − x) = Σ xⁿ (|x| < 1) - Ratio test:
R = lim |aₙ/aₙ₊₁| - Frobenius:
y = Σ aₙ x^(n + r) near a regular singular point
Worked example: Recurrence for y″ + y = 0
Given: y = Σ aₙxⁿ
Find: Recurrence and solution.
- Σ (n + 2)(n + 1)aₙ₊₂xⁿ + Σ aₙxⁿ = 0
- aₙ₊₂ = −aₙ/((n + 2)(n + 1))
- a₀ series → cos x; a₁ series → sin x
Answer: y = a₀ cos x + a₁ sin x
Worked example: Taylor estimate
Given: e^0.1 using 3 terms
Find: Approximation and error.
- 1 + 0.1 + 0.005 = 1.105
- Exact 1.105171
Answer: Error ≈ 1.7 × 10⁻⁴ (≈ next term 0.1³/6)
Common mistakes
- Wrong: Using a series outside its radius of convergence. — Right: Check R first.
- Wrong: Misaligning indices in recurrences. — Right: Shift so every sum has the same power.
Applications
- Small-signal models: Linear and quadratic approximations.
- Special functions: Bessel and Legendre solutions.
- Numerical libraries: Series evaluate sin, exp in software.
Key takeaways
- Taylor: Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n!.
- Check radius of convergence.
- Power series → recurrence relations.
- Frobenius at regular singular points.
Going further
- Asymptotic series: Divergent but useful approximations (M24).
- Padé approximants: Rational approximations beyond Taylor.
- Analytic continuation: Extending series beyond R.