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MathidiaMathematics Academy
Differential equations II
Lesson 28 of 75
Standard practice

Series solutions

Objective. Construct power-series solutions near ordinary points.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Series solutions express the solution of a linear ODE as a power series Σ aₙ(x−x0)ⁿ about an ordinary point x0, where the coefficients of the equation are analytic. Substituting the series into the ODE and matching coefficients of each power of x produces a recurrence relation for aₙ, with a0 and a1 free to satisfy initial conditions. For y″ − y = 0 about x = 0, matching powers gives aₙ₊2 = aₙ/((n+1)(n+2)), reproducing the familiar e^x and e^(−x) series. The method extends to equations without elementary closed-form solutions.

Worked example

Find the recurrence relation for the series solution of y″ = y about the ordinary point x = 0.

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  1. 1Write y = Σ aₙxⁿ so y″ = Σ n(n−1)aₙx^(n−2)
  2. 2Shift index: y″ = Σ (n+2)(n+1)aₙ₊2 xⁿ
  3. 3Equate coefficients with y = Σ aₙxⁿ term by term
  4. 4Solve for aₙ₊2 in terms of aₙ

Answer: aₙ₊2 = aₙ/((n+1)(n+2))

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Common mistake. A frequent mistake is shifting the summation index incorrectly when combining series with different starting powers, which misaligns the recurrence. Always re-index every sum to start at the same power of x before equating coefficients.

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