Questions: Power Series Solutions to Differential Equations

Question 1

After substituting y = Σ aₙxⁿ into a second-order ODE and collecting terms, you find the recurrence aₙ₊₂ = −aₙ / [(n+2)(n+1)]. How many free parameters are there, and how many linearly independent solutions does this produce?

• A. One free parameter (a₀); the recurrence generates a single solution
• B. Two free parameters (a₀ and a₁); choosing (1,0) and (0,1) generates two independent solutions
• C. Three free parameters (a₀, a₁, a₂); this is a third-order recurrence
• D. All coefficients are determined once a₀ is known; there is only one solution

Answer: (B)

A second-order ODE has a two-dimensional solution space, which corresponds to two free parameters. The recurrence aₙ₊₂ = −aₙ/[(n+2)(n+1)] links even-indexed coefficients (a₀, a₂, a₄, …) and odd-indexed coefficients (a₁, a₃, a₅, …) independently — so a₀ freely determines all even terms, and a₁ freely determines all odd terms. Setting (a₀,a₁) = (1,0) gives one solution; (0,1) gives a second. These are linearly independent (neither is a multiple of the other) and their span is the general solution. Option D is the most common error: forgetting that a₁ is also free.

Question 2

A student applies the power series method to an ODE and obtains a recurrence relation. After computing 20 terms, they cannot recognize the resulting series as any standard function (e₊, cos, etc.). They conclude the method has failed. What is wrong with this reasoning?

• A. The method only works if the solution is a polynomial; an unrecognized series means a singular point was encountered
• B. The power series method produces a valid solution even when the series cannot be expressed in closed form — the series itself is the answer and can be truncated for approximation
• C. An unrecognized series means the recurrence contains an error and must be recomputed
• D. The student should switch to the Frobenius method whenever the solution is not immediately recognizable

Answer: (B)

Many important ODEs — Bessel's equation, Legendre's equation, and countless others in physics — have solutions that cannot be expressed as finite combinations of elementary functions. The power series method still succeeds in these cases: the series IS the solution. This is exactly how 'special functions' like Bessel functions were originally defined — as named power series that solve specific ODEs. Truncating the series to a finite number of terms gives arbitrarily accurate numerical approximations. The method fails only when the expansion point is singular in a way that requires the Frobenius modification.

Question 3

The power series method can be applied to any second-order ODE, regardless of whether the expansion point is ordinary or singular.

• T. True
• F. False

Answer: False

The standard power series method (assuming y = Σ aₙxⁿ with integer powers) works reliably at ordinary points, where the coefficient functions are analytic. At a singular point, the solutions may involve logarithms or non-integer powers of x, which the standard power series ansatz cannot represent. Applying the method blindly at a singular point may yield only one solution or no valid solution at all. The Frobenius method — which assumes y = xʳ Σ aₙxⁿ for some (possibly non-integer) exponent r — is the appropriate tool at regular singular points.

Question 4

Setting a₀ = 1, a₁ = 0 and then a₀ = 0, a₁ = 1 generates two linearly independent power series solutions to a second-order ODE.

• T. True
• F. False

Answer: True

Because the recurrence relation links every coefficient back to a₀ and a₁, specifying these two values determines the entire series. The series with (a₀,a₁) = (1,0) contains only even-powered terms (in recurrences that couple even and odd indices separately); the series with (0,1) contains only odd-powered terms. These are linearly independent: neither is a scalar multiple of the other (one has nonzero even terms, the other has nonzero odd terms). Together they span the full solution space. This is the power series analog of the two linearly independent solutions guaranteed by the theory of second-order linear ODEs.

Question 5

Why does substituting a power series into an ODE produce infinitely many algebraic equations, and what makes this infinite system tractable?

Model answer: Substituting y = Σ aₙxⁿ into the ODE and simplifying produces a new power series that must equal zero. A power series equals zero if and only if every coefficient is independently zero — one condition per power of x, hence infinitely many equations. The system is tractable because these equations are not independent: they are linked by a recurrence relation (e.g., aₙ₊₂ depends only on aₙ). Instead of solving an infinite system of unrelated equations, you identify one recurrence and use it to compute any coefficient from two free initial values (a₀ and a₁). The infinite complexity collapses into a single repeating rule.

This is the core insight distinguishing the power series method from naive coefficient-matching. The recurrence structure means you never actually solve an infinite system; you solve a simple recursive formula once and propagate it. The infinite series emerges automatically from two initial choices.