Power series solutions near ordinary points
When an equation has variable coefficients and no elementary solution, we can still build the solution term by term as a power series.
The idea
Assume \(y = \sum_{n=0}^{\infty} a_nx^n\), substitute into the equation, and match coefficients of each power of \(x\). This produces a recurrence relation for the \(a_n\). The first two coefficients, \(a_0 = y(0)\) and \(a_1 = y'(0)\), are free, matching the two constants of a second-order equation.
y = \sum_{n=0}^{\infty} a_n x^nOrdinary and singular points
For \(y'' + p(x)y' + q(x)y = 0\), a point \(x_0\) is ordinary if \(p\) and \(q\) are analytic (have convergent power series) there. Near ordinary points, two independent power series solutions exist. Points where \(p\) or \(q\) blow up are singular points and need special methods.
Worked example: y'' + y = 0
Substituting \(y = \sum a_nx^n\) gives \(\sum [(n+2)(n+1)a_{n+2} + a_n]x^n = 0\), so \(a_{n+2} = -\frac{a_n}{(n+2)(n+1)}\). Starting from \(a_0\): \(a_2 = -\frac{a_0}{2}\), \(a_4 = \frac{a_0}{24}\), and so on, which is \(a_0\cos x\). Starting from \(a_1\) gives \(a_1\sin x\). The method recovers the familiar solutions.
a_{n+2} = -\frac{a_n}{(n+2)(n+1)}Shifting indices
Matching coefficients requires every sum to use the same power. If \(y'' = \sum n(n-1)a_nx^{n-2}\), substitute \(k = n - 2\) to get \(\sum (k+2)(k+1)a_{k+2}x^k\). Careful index shifting is the main technical skill in series solutions.
Airy's equation
For \(y'' - xy = 0\), the recurrence is \(a_{n+2} = \frac{a_{n-1}}{(n+2)(n+1)}\) with \(a_2 = 0\). The solutions are Airy functions, which describe light near a rainbow's edge and quantum particles in a linear potential. They have no formula in elementary functions, but the series defines them precisely.
Radius of convergence
A power series solution about an ordinary point converges at least out to the nearest singular point of the equation in the complex plane. For Legendre's equation \((1 - x^2)y'' - 2xy' + n(n+1)y = 0\), the singular points \(\pm 1\) give radius at least 1 about \(x = 0\).
Legendre polynomials
When \(n\) is a non-negative integer, one series solution of Legendre's equation terminates, giving a polynomial: \(P_0 = 1\), \(P_1 = x\), \(P_2 = \frac{1}{2}(3x^2 - 1)\). Legendre polynomials appear in gravitational and electric potentials and in the shapes of atomic orbitals.
Common mistakes
Mismatching powers when combining sums. Forgetting that \(a_0\) and \(a_1\) are free. Assuming the series converges everywhere. Losing track of the lowest terms that do not fit the general recurrence.
Key takeaways
Substitute a power series, shift indices to align powers, and match coefficients to get a recurrence. Near ordinary points, two independent series solutions exist with \(a_0\) and \(a_1\) free. Convergence extends at least to the nearest singular point. Airy functions and Legendre polynomials arise this way.
Flashcard
y = \sum a_nx^n
Check your understanding
In a power series solution of a second-order equation about an ordinary point, which coefficients are arbitrary?