27-07 — Series Solutions and Frobenius Method

Phase: 27 — Ordinary Differential Equations | Subject: 27-07 Prerequisites: 27-03 (Second-Order Linear ODEs), 03-07 (Sequences and Series) Next subject: 27-08 — Numerical Methods for ODEs

Learning Objectives

By the end of this subject, you will be able to:

1. Understand when series solutions are needed (variable-coefficient ODEs)
2. Apply the power series method to solve ODEs around ordinary points
3. Use the Frobenius method for regular singular points
4. Derive and work with Bessel functions as classic series solutions
5. Determine the radius of convergence for series solutions

Core Content

1. Why Series Solutions?

When an ODE has variable coefficients (coefficients that are functions of x), standard methods (characteristic equation, integrating factor) often fail. Series solutions express the solution as a power series:

$$y = \sum_{n=0}^{\infty} a_n x^n$$

This turns the ODE into a recurrence relation for the coefficients $a_n$.

2. Power Series Method (Ordinary Points)

An ordinary point $x_0$ is where $P(x) \neq 0$ for a standard-form ODE $P(x)y'' + Q(x)y' + R(x)y = 0$.

Method: 1. Assume $y = \sum_{n=0}^{\infty} a_n (x-x_0)^n$ 2. Compute $y'$ and $y''$ term-by-term 3. Substitute into the ODE 4. Collect like powers of $(x-x_0)$ 5. Set the coefficient of each power to 0 (gives recurrence) 6. Solve for $a_n$ recursively

3. Frobenius Method (Regular Singular Points)

A regular singular point occurs where $P(x_0) = 0$ but $(x-x_0)Q(x)/P(x)$ and $(x-x_0)^2 R(x)/P(x)$ remain finite.

Frobenius form: $$y = \sum_{n=0}^{\infty} a_n x^{n+r}$$

where $r$ is the indicial exponent found from the indicial equation.

4. Bessel's Equation

The canonical example: $x^2 y'' + x y' + (x^2 - \nu^2)y = 0$

Solutions are Bessel functions $J_\nu(x)$ and $Y_\nu(x)$ (Bessel functions of the first and second kind).

5. Radius of Convergence

A power series solution converges within a radius $R$ determined by the nearest singularity of the ODE's coefficients. Within this radius, the series solution equals the true solution.

Worked Examples

Example 1: Power Series for a Simple ODE

Problem: Solve $y'' + y = 0$ using a power series around $x_0 = 0$.

Solution:

Assume $y = \sum_{n=0}^{\infty} a_n x^n$.

Then $y' = \sum_{n=1}^{\infty} n a_n x^{n-1}$ and $y'' = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}$.

Substituting: $\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + \sum_{n=0}^{\infty} a_n x^n = 0$

Shift indices to match powers: $\sum_{n=0}^{\infty} [(n+2)(n+1) a_{n+2} + a_n] x^n = 0$

Recurrence: $(n+2)(n+1) a_{n+2} + a_n = 0$, so $a_{n+2} = -a_n / [(n+2)(n+1)]$

• $a_2 = -a_0/2$
• $a_4 = -a_2/12 = a_0/4!$
• $a_6 = -a_4/30 = -a_0/6!$

Even terms: $a_{2n} = (-1)^n a_0 / (2n)!$

Similarly for odd: $a_{2n+1} = (-1)^n a_1 / (2n+1)!$

Answer: $y = a_0 \cos(x) + a_1 \sin(x)$ — recovers the known solution via series.

Example 2: Frobenius Method

Problem: Find the indicial equation for Bessel's equation: $x^2 y'' + x y' + (x^2 - \nu^2)y = 0$.

Solution:

Assume $y = \sum_{n=0}^{\infty} a_n x^{n+r}$.

Compute derivatives and substitute. The lowest power is $x^r$:

Coefficient of $x^r$: $[r(r-1) + r - \nu^2] a_0 = [r^2 - \nu^2] a_0 = 0$

Indicial equation: $r^2 - \nu^2 = 0$, so $r = \pm \nu$.

Answer: The indicial exponents are $r = \nu$ and $r = -\nu$.

Example 3: Radius of Convergence

Problem: For the ODE $y'' + (1+x)y = 0$, determine the radius of convergence of the series solution about $x_0 = 0$.

Solution:

Standard form: $P(x) = 1$, $Q(x) = 1+x$, $R(x) = 1+x$.

All coefficients are polynomials — analytic everywhere. The nearest singularity is at infinity.

Answer: Radius of convergence $R = \infty$ (series converges for all $x$).

Practice Problems

Problem 1: Use power series to solve $y'' - 2xy' + y = 0$ (Hermite's equation). Find the first 4 non-zero terms.

Problem 2: For the ODE $(1-x^2)y'' - 2xy' + 2y = 0$ (Legendre's equation), find the indicial equation and the first two terms of each solution.

Problem 3: Show that the series solution for $y' = xy$ with $y(0) = 1$ gives $y = e^{x^2/2}$.

Problem 4: For Bessel's equation with $\nu = 0$, write out the first 4 terms of $J_0(x)$.

Problem 5: Explain why the Frobenius method is needed for $xy'' + y = 0$ but not for $y'' + y = 0$.

Summary

1. Series solutions express ODE solutions as power series when standard methods fail
2. Power series method works at ordinary points by substituting a Taylor series into the ODE
3. Frobenius method extends this to regular singular points using $y = \sum a_n x^{n+r}$
4. Bessel functions arise from series solutions of Bessel's equation and appear throughout physics
5. Radius of convergence is determined by the nearest singularity of the ODE coefficients

Pitfalls

• Not shifting indices correctly when substituting series into the ODE: When $y = \sum a_n x^n$, then $y' = \sum n a_n x^{n-1}$ and $y'' = \sum n(n-1)a_n x^{n-2}$. All sums must be expressed in terms of the same power of $x$ (typically $x^n$) by shifting indices. A frequent error: writing $y'' = \sum n(n-1)a_n x^{n-2}$ but then treating it as $\sum n(n-1)a_n x^n$, resulting in coefficients that are off by two index positions and a completely wrong recurrence.
• Misidentifying singular points: A point $x_0$ is ordinary if both $P(x)$ and $Q(x)$ in $y'' + P(x)y' + Q(x)y = 0$ are analytic at $x_0$. It is a regular singular point if $(x-x_0)P(x)$ and $(x-x_0)^2 Q(x)$ are analytic. A common mistake: declaring a point "regular singular" just because $P$ or $Q$ blows up, without checking the multiplicative factors. For Bessel's equation $x^2 y'' + x y' + (x^2 - \nu^2)y = 0$, divide through by $x^2$ to get $P(x) = 1/x$ and $Q(x) = 1 - \nu^2/x^2$. Then $xP(x) = 1$ and $x^2 Q(x) = x^2 - \nu^2$, both analytic → regular singular at $x=0$.
• Forming the indicial equation incorrectly: The indicial equation comes from the lowest-power term after substituting $y = \sum a_n x^{n+r}$. For $x^2 y'' + x p(x) y' + q(x) y = 0$, it's $r(r-1) + p(0)r + q(0) = 0$. A common error: using $p(x)$ and $q(x)$ directly instead of their constant terms $p(0)$ and $q(0)$, or forgetting that $p(x)$ and $q(x)$ come from rewriting in standard form.
• Assuming the radius of convergence is always the distance to the nearest real singular point: Singularities can be complex. For $(1+x^2)y'' + y = 0$, the singular points are $x = \pm i$, not real. The radius of convergence about $x_0 = 0$ is $|i| = 1$, even though the equation "looks fine" for all real $x$. Always search the complex plane for singularities of $P(x)$ and $Q(x)$.
• Misunderstanding the logarithmic case in Frobenius: When the indicial roots differ by an integer, the second solution may contain $\ln x$ — but not always. For Bessel's equation with $\nu = 0$, $r_1 = r_2 = 0$, so the second solution $Y_0(x)$ contains $\ln x$. But for $\nu = 1/2$, the roots differ by 1 (an integer), yet $J_{-1/2}(x) = \sqrt{2/(\pi x)}\cos x$ has no logarithm. Always check the recurrence: if it produces a second independent series without $\ln x$, you don't need it.

Key Terms

• Bessel functions
• Frobenius method
• Indicial equation
• Indicial exponent
• Ordinary point
• Power series method
• Radius of convergence
• Regular singular point
• Recurrence relation
• Series solution

Quiz

Q1: When is a power series solution method needed for an ODE?

A) Only for first-order ODEs B) When the ODE has variable coefficients that prevent standard methods C) Only for nonlinear ODEs D) When the initial conditions are missing

Correct: B)

• If you chose B: Correct. Power series methods handle variable-coefficient ODEs where characteristic equations or integrating factors fail.
• If you chose A: First-order ODEs often have simpler solutions. Series methods are most valuable for higher-order ODEs with variable coefficients.
• If you chose C: Non-linearity is a separate challenge. Series methods can help with some nonlinear ODEs, but the primary use case is linear ODEs with variable coefficients.
• If you chose D: Missing initial conditions affect uniqueness, not whether a series solution exists.

Q2: In the Frobenius method, the form of the assumed solution is:

A) $y = \sum a_n x^n$ B) $y = \sum a_n x^{n+r}$ where $r$ is the indicial exponent C) $y = \sum a_n (x-x_0)^n$ for any $x_0$ D) $y = \sum a_n \sin(nx)$

Correct: B)

• If you chose B: Correct. The Frobenius method assumes $y = \sum a_n x^{n+r}$ to handle regular singular points where the power series alone is insufficient.
• If you chose A: That's the standard power series method (ordinary points), not Frobenius.
• If you chose C: This is the Taylor series form. Frobenius specifically uses $x^{n+r}$ with a fractional shift.
• If you chose D: Trigonometric series are Fourier series, not used for Frobenius solutions.

Q3: The indicial equation determines:

A) The convergence radius of the series B) The possible values of the exponent $r$ in the Frobenius solution C) The initial conditions for the ODE D) Whether the ODE is linear or nonlinear

Correct: B)

• If you chose B: Correct. The indicial equation is derived from the lowest power of $x$ in the substituted series and gives the allowed values of $r$.
• If you chose A: The radius of convergence comes from the nearest singularity, not the indicial equation.
• If you chose C: Initial conditions determine $a_0, a_1$ after finding the series form, not the indicial equation.
• If you chose D: Linearity is determined by the ODE structure, not the indicial equation.

Q4: Bessel functions $J_\nu(x)$ arise as solutions to:

A) $y'' + \lambda y = 0$ B) $x^2 y'' + x y' + (x^2 - \nu^2)y = 0$ C) $y'' + p(x)y' + q(x)y = 0$ for any $p, q$ D) $y' = ky$

Correct: B)

• If you chose B: Correct. This is Bessel's equation, whose solutions are the Bessel functions.
• If you chose A: That's the simple harmonic oscillator equation, giving sin/cos solutions.
• If you chose C: That's the general linear ODE. Bessel functions come from the specific equation in B.
• If you chose D: That's a separable first-order ODE with exponential solution.

Q5: The radius of convergence for a series solution about an ordinary point is:

A) Always 1 B) Always 0 C) The distance to the nearest singularity of the ODE coefficients D) Infinite for all ODEs

Correct: C)

• If you chose C: Correct. The radius of convergence equals the distance from the expansion point to the nearest singularity in the complex plane.
• If you chose A: Only true if the nearest singularity is at distance 1. Generally false.
• If you chose B: A radius of 0 means the series never converges, which is not typical for ODE solutions.
• If you chose D: Only true if all coefficients are entire functions (analytic everywhere). Not generally true.

Q6: In the series solution method, substituting the power series into the ODE and equating coefficients of each power of $x$ to zero yields:

A) The general solution directly B) A recurrence relation for the coefficients $a_n$ C) The initial conditions D) The characteristic equation

Correct: B)

• If you chose B: Correct. Equating coefficients gives a recurrence relation $a_{n+2} = f(n, a_n, a_{n-1}, ...)$ that determines all coefficients.
• If you chose A: You still need to solve the recurrence and identify the pattern. The general solution emerges after solving.
• If you chose C: Initial conditions determine the specific values of $a_0, a_1$, but the recurrence gives the general form.
• If you chose D: The characteristic equation is for constant-coefficient ODEs, not series solutions.

Q7: For the ODE $y'' + xy' + y = 0$ about $x_0 = 0$, the recurrence relation for the coefficients is:

A) $a_{n+2} = -a_n / [(n+2)(n+1)]$ B) $a_{n+2} = -(a_n + n a_{n-1}) / [(n+2)(n+1)]$ C) $a_{n+2} = a_n / (n+2)$ D) No recurrence exists — this ODE cannot be solved by series

Correct: B)

• If you chose B: Correct. The $xy'$ term introduces an $n a_{n-1}$ contribution, giving the more complex recurrence.
• If you chose A: That's the recurrence for $y'' + y = 0$ (no first-derivative term). The extra $xy'$ changes the recurrence.
• If you chose C: This recurrence is too simple and doesn't match the ODE structure.
• If you chose D: This ODE is solvable by power series. The presence of $xy'$ makes it require a series approach.

Q8: Frobenius method is specifically designed for ODEs with:

A) Constant coefficients B) Nonlinear terms C) Regular singular points D) Periodic coefficients

Correct: C)

• If you chose C: Correct. The Frobenius method extends power series solutions to handle regular singular points where $P(x_0) = 0$ but related functions remain finite.
• If you chose A: Constant-coefficient ODEs are solved by characteristic equations, not series methods.
• If you chose B: Non-linearity is a separate challenge. Frobenius applies to linear ODEs with singular points.
• If you chose D: Periodic coefficients are handled by Floquet theory, not Frobenius.

Next Steps

Continue to 27-08 — Numerical Methods for ODEs to learn how computers solve ODEs when analytical solutions are impossible — including Euler's method, Runge-Kutta, and stability analysis.