Series Solutions

Learning Objectives

  • Understand how to represent solutions to differential equations as power series.
  • Determine the radius of convergence using the Ratio Test.
  • Classify ordinary and singular points for variable-coefficient DEs.
  • Apply the method of power series around an ordinary point.
  • Apply the Frobenius method around a regular singular point.

When a differential equation cannot be solved by elementary analytical methods (especially when it involves variable coefficients like x2y′′+y=0x^2 y'' + y = 0), we often assume the solution can be represented as an infinite power series.

Power Series Solution

We assume a solution of the form of an infinite series. Typically, we expand the series around x0=0x_0 = 0 (a Maclaurin series).

Power Series Solution Form

The general assumed form of a power series solution.

y(x)=∑n=0∞cn(x−x0)ny(x) = \sum_{n=0}^{\infty} c_n (x-x_0)^n

Variables

SymbolDescriptionUnit
cnc_nThe constant coefficients of the series-
x0x_0The point around which the series is expanded-

Maclaurin Series Solution Form

The assumed form of a power series expanded around x0=0x_0 = 0.

y(x)=∑n=0∞cnxn=c0+c1x+c2x2+c3x3+…y(x) = \sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \dots

Variables

SymbolDescriptionUnit
cnc_nThe constant coefficients of the series-

Convergence and the Ratio Test

A power series is only useful if it converges to a finite value. The interval of xx values for which the series converges is called the interval of convergence, centered at x0x_0, and half its length is the radius of convergence, RR.

Finding the Radius of Convergence

For ∑n=0∞cn(x−x0)n\sum_{n=0}^{\infty}c_n(x-x_0)^n, the root test gives the radius

1R=lim sup⁡n→∞∣cn∣1/n.\frac{1}{R}=\limsup_{n\to\infty}|c_n|^{1/n}.

When the successive-coefficient ratio has a limit, the ratio test gives the same radius. For an individual xx, a term-ratio limit less than 1 gives absolute convergence, greater than 1 gives divergence, and equal to 1 is inconclusive. Test each real endpoint separately. If R=0R=0, only the center is guaranteed by the radius; if R=∞R=\infty, the power series converges for every finite xx.

Ratio Test for Power Series

Use the ratio of successive terms at a chosen x: a limit below 1 gives absolute convergence, above 1 gives divergence, and equal to 1 is inconclusive.

L(x)=lim⁡n→∞∣an+1(x)an(x)∣L(x) = \lim_{n \to \infty} \left| \frac{a_{n+1}(x)}{a_n(x)} \right|

Variables

SymbolDescriptionUnit
ana_nThe n-th term of the series-
L(x)L(x)The successive-term ratio limit at the chosen x; equality to 1 is inconclusive-
The Radius of a Local Power Series

Coefficient growth determines the series radius; convergence inside the radius and convergence at each endpoint are separate questions.

Center the series at x₀
Coefficient growth determines radius R
Absolute convergence for |x−x₀| < R
Check each endpoint separately; the ratio test may be inconclusive there

Power Series Convergence

Before diving into solving DEs with series, it helps to visualize how a power series converges to a target function as more terms are added within its radius of convergence. This is the foundation of Taylor series approximations.

Interactive Simulation

Use the simulation below to explore how power series solutions converge as terms are added.

Power Series Solution

Solving y′=yy' = y with y(0)=1y(0)=1 around the ordinary point x0=0x_0=0.

y(x)≈∑n=00xnn!y(x) \approx \sum_{n=0}^{0} \frac{x^n}{n!}

Notice how adding more terms increases the interval where the polynomial approximation closely matches the exact exponential solution.

1
LinearDegree 9
Exact Solution exe^x
Series Approximation
x ∈ [-3, 3]

Ordinary vs. Singular Points

Before solving a second-order linear equation A2(x)y′′+A1(x)y′+A0(x)y=0A_2(x)y'' + A_1(x)y' + A_0(x)y = 0, normalize it on a neighborhood of the expansion point and classify that point using the resulting coefficients p(x)p(x) and q(x)q(x).

Classification of Points

On a punctured neighborhood when necessary, rewrite the equation in standard form y′′+p(x)y′+q(x)y=0y'' + p(x)y' + q(x)y = 0, where p(x)=A1(x)/A2(x)p(x) = A_1(x)/A_2(x) and q(x)=A0(x)/A2(x)q(x) = A_0(x)/A_2(x). Assume the original coefficients are analytic on the neighborhood.

  • Ordinary Point: Both pp and qq are analytic at x0x_0. When the original coefficients are analytic and the equation is written in reduced standard form, a nonzero leading coefficient at x0x_0 is a sufficient check. Use an ordinary Taylor series.

  • Singular Point: At least one normalized coefficient is not analytic at x0x_0.

    • Regular Singular Point: (x−x0)p(x)(x-x_0)p(x) and (x−x0)2q(x)(x-x_0)^2q(x) are both analytic at x0x_0. The Frobenius method is the standard local series method.
    • Irregular Singular Point: At least one regular-singular condition fails. Ordinary Taylor and standard Frobenius methods are not generally guaranteed.
Ordinary and Regular-Singular Points Use Different Series Bases

The point classification changes the assumed local powers: Taylor powers begin at zero, while the Frobenius form includes an indicial exponent.

Ordinary point
  • p and q analytic at x₀
  • Assume Σ cₙ(x−x₀)ⁿ
  • Determine coefficients from a recurrence
Regular singular point
  • (x−x₀)p and (x−x₀)²q analytic
  • Assume (x−x₀)^r Σ cₙ(x−x₀)ⁿ
  • Find indicial roots before the recurrence

Method of Power Series (Around an Ordinary Point)

  1. Assume Solution: Write y=∑n=0∞cn(x−x0)ny = \sum_{n=0}^\infty c_n (x-x_0)^n.
  2. Differentiate: Differentiate the series term by term within its interval of convergence.
  3. Substitute: Plug y,y′,y′′y, y', y'' into the DE.
  4. Shift Indices: Reindex every sum so like powers of (x−x0)(x-x_0) have matching exponents.
  5. Find Recurrence Relation: Set each power-series coefficient to zero. The recurrence and its index pattern depend on the differential equation.
  6. Check: Determine independent solution constants, convergence domain, and any endpoint behavior required by the problem.
How Power-Series Recurrences Are Assembled

Differentiation shifts exponents; after reindexing, each coefficient equation combines only terms with the same power of x−x₀.

Assume y = Σ cₙ(x−x₀)ⁿ
Differentiate to form y′ and y″
Reindex every term to a common power
Set each common-power coefficient to zero
Solve the recurrence for the series coefficients

Method of Frobenius (Around a Regular Singular Point)

If x0x_0 is a regular singular point, an ordinary Taylor series is not the standard guaranteed form. Use a Frobenius series y=∑n=0∞cn(x−x0)n+ry = \sum_{n=0}^{\infty} c_n(x-x_0)^{n+r} with c0≠0c_0 \neq 0 and determine the exponent rr from the indicial equation.

Frobenius Series Solution

The assumed solution form around a regular singular point.

y(x)=∑n=0∞cn(x−x0)n+r=(x−x0)r∑n=0∞cn(x−x0)ny(x) = \sum_{n=0}^{\infty} c_n (x-x_0)^{n+r} = (x-x_0)^r \sum_{n=0}^{\infty} c_n (x-x_0)^n

Variables

SymbolDescriptionUnit
cnc_nThe constant coefficients of the series-
rrThe indicial root-

Applying the Method of Frobenius

  1. Assume Solution: Write y=∑n=0∞cn(x−x0)n+ry = \sum_{n=0}^{\infty} c_n (x-x_0)^{n+r}.
  2. Find Derivatives: Differentiate the assumed series term by term where it converges.
  3. Substitute into DE: Substitute into the differential equation, distribute any xx coefficients, and shift indices to align the powers of xx.
  4. Indicial Equation: Set the coefficient of the lowest power to zero. Since c0≠0c_0 \neq 0, this yields the indicial equation for rr.
  5. Solve for r: Find indicial roots r1,r2r_1,r_2. If they repeat or differ by an integer, use the corresponding root-case method; the second independent solution may contain a logarithm.
  6. Verify: Derive the recurrence, establish independent solutions, and determine the convergence domain.

Choose the Local Series Method

Power-Series, Frobenius, or Alternative Method

Classify the expansion point of a normalized second-order linear ODE and choose the supported local solution method.

Write y″ + p(x)y′ + q(x)y = 0 on a neighborhood of x₀, allowing p and q to be defined on a punctured neighborhood when x₀ is singular. An ordinary point has p and q analytic at x₀. A regular singular point has (x−x₀)p and (x−x₀)²q analytic at x₀. Other singular points are irregular. At a regular singular point the indicial-root relation matters: when roots repeat or differ by an integer, the second independent solution can require a logarithmic term and must be handled by the corresponding root-case method. A local series must still satisfy the recurrence, independence, interval/domain, and convergence requirements; the ratio test is inconclusive when its limit equals 1, and endpoints need separate checks.

Power-Series, Frobenius, or Alternative MethodClassify the expansion point of a normalized second-order linear ODE and choose the supported local solution method.. Write y″ + p(x)y′ + q(x)y = 0 on a neighborhood of x₀, allowing p and q to be defined on a punctured neighborhood when x₀ is singular. An ordinary point has p and q analytic at x₀. A regular singular point has (x−x₀)p and (x−x₀)²q analytic at x₀. Other singular points are irregular. At a regular singular point the indicial-root relation matters: when roots repeat or differ by an integer, the second independent solution can require a logarithmic term and must be handled by the corresponding root-case method. A local series must still satisfy the recurrence, independence, interval/domain, and convergence requirements; the ratio test is inconclusive when its limit equals 1, and endpoints need separate checks.. Start: choose expansion point x₀ → Normalize to y″ + p(x)y′ + q(x)y = 0; Normalize to y″ + p(x)y′ + q(x)y = 0 → Are p and q analytic at x₀?; Are p and q analytic at x₀? — Yes → Assume a Taylor series and derive its recurrence; Are p and q analytic at x₀? — No → Are (x−x₀)p and (x−x₀)²q analytic?; Assume a Taylor series and derive its recurrence → Check independence, convergence, and data; Are (x−x₀)p and (x−x₀)²q analytic? — Yes → Assume y = (x−x₀)^r Σ cₙ(x−x₀)ⁿ; Are (x−x₀)p and (x−x₀)²q analytic? — No → Use an appropriate alternative or asymptotic method; Assume y = (x−x₀)^r Σ cₙ(x−x₀)ⁿ → Find indicial equation and its roots; Find indicial equation and its roots → Repeated roots or integer difference?; Repeated roots or integer difference? — Yes → Apply the root-case rule; a log term may occur; Repeated roots or integer difference? — No → Construct both Frobenius solutions; Apply the root-case rule; a log term may occur → Derive and solve coefficient recurrence; Construct both Frobenius solutions → Derive and solve coefficient recurrence; Derive and solve coefficient recurrence → Check independence, convergence, and data; Check independence, convergence, and data → Series satisfies the ODE and conditions?; Series satisfies the ODE and conditions? — Yes → Record local solution and validity domain; Series satisfies the ODE and conditions? — No → Recheck index shifts, roots, and convergence domain; Recheck index shifts, roots, and convergence domain → Normalize to y″ + p(x)y′ + q(x)y = 0; Use an appropriate alternative or asymptotic method → End chart and use selected method

Start: choose expansion point x₀ → Normalize to y″ + p(x)y′ + q(x)y = 0; Normalize to y″ + p(x)y′ + q(x)y = 0 → Are p and q analytic at x₀?; Are p and q analytic at x₀? — Yes → Assume a Taylor series and derive its recurrence; Are p and q analytic at x₀? — No → Are (x−x₀)p and (x−x₀)²q analytic?; Assume a Taylor series and derive its recurrence → Check independence, convergence, and data; Are (x−x₀)p and (x−x₀)²q analytic? — Yes → Assume y = (x−x₀)^r Σ cₙ(x−x₀)ⁿ; Are (x−x₀)p and (x−x₀)²q analytic? — No → Use an appropriate alternative or asymptotic method; Assume y = (x−x₀)^r Σ cₙ(x−x₀)ⁿ → Find indicial equation and its roots; Find indicial equation and its roots → Repeated roots or integer difference?; Repeated roots or integer difference? — Yes → Apply the root-case rule; a log term may occur; Repeated roots or integer difference? — No → Construct both Frobenius solutions; Apply the root-case rule; a log term may occur → Derive and solve coefficient recurrence; Construct both Frobenius solutions → Derive and solve coefficient recurrence; Derive and solve coefficient recurrence → Check independence, convergence, and data; Check independence, convergence, and data → Series satisfies the ODE and conditions?; Series satisfies the ODE and conditions? — Yes → Record local solution and validity domain; Series satisfies the ODE and conditions? — No → Recheck index shifts, roots, and convergence domain; Recheck index shifts, roots, and convergence domain → Normalize to y″ + p(x)y′ + q(x)y = 0; Use an appropriate alternative or asymptotic method → End chart and use selected method

  • Start: choose expansion point x₀: terminator
  • Normalize to y″ + p(x)y′ + q(x)y = 0: process
  • Are p and q analytic at x₀?: decision
  • Assume a Taylor series and derive its recurrence: process
  • Are (x−x₀)p and (x−x₀)²q analytic?: decision
  • Assume y = (x−x₀)^r Σ cₙ(x−x₀)ⁿ: process
  • Find indicial equation and its roots: process
  • Repeated roots or integer difference?: decision
  • Apply the root-case rule; a log term may occur: process
  • Construct both Frobenius solutions: process
  • Derive and solve coefficient recurrence: process
  • Check independence, convergence, and data: process
  • Series satisfies the ODE and conditions?: decision
  • Recheck index shifts, roots, and convergence domain: process
  • Use an appropriate alternative or asymptotic method: process
  • Record local solution and validity domain: terminator
  • End chart and use selected method: terminator
Key Takeaways
  • Power Series Method: Assume y=∑cn(x−x0)ny = \sum c_n (x-x_0)^n around an Ordinary Point, where the normalized coefficients pp and qq are analytic.
  • Radius of Convergence (RR): Use the ratio/root test to determine RR; check endpoints separately because the interior radius alone does not decide endpoint convergence.
  • Recurrence Relation: The core result of the method. It defines how to calculate higher-order coefficients cnc_n based on previous ones, usually leaving c0c_0 and c1c_1 as arbitrary constants.
  • Frobenius Method: Use the form y=(x−x0)r∑cn(x−x0)ny = (x-x_0)^r \sum c_n (x-x_0)^n at a Regular Singular Point, then solve the indicial equation and handle integer-difference root cases carefully.

References