The Frobenius method and regular singular points
Near some singular points, solutions behave like a power of x times a power series. The Frobenius method finds them.
Regular singular points
A singular point \(x_0 = 0\) of \(y'' + p(x)y' + q(x)y = 0\) is regular if \(xp(x)\) and \(x^2q(x)\) are analytic there. Then solutions can still be found by a modified series. Cauchy-Euler equations are the simplest example.
The Frobenius ansatz
Try \(y = x^r\sum_{n=0}^{\infty} a_nx^n\) with \(a_0 \ne 0\). The lowest-power terms give the indicial equation for \(r\), and higher powers give a recurrence for the \(a_n\). The exponent \(r\) need not be an integer, allowing solutions like \(\sqrt{x}\) times a series.
y = x^r \sum_{n=0}^{\infty} a_n x^nThe indicial equation
If \(xp(x) \to p_0\) and \(x^2q(x) \to q_0\) as \(x \to 0\), the indicial equation is \(r(r - 1) + p_0r + q_0 = 0\). Its roots \(r_1 \ge r_2\) determine the leading behaviour of the solutions near the singular point.
r(r-1) + p_0 r + q_0 = 0Cases for the roots
If \(r_1 - r_2\) is not an integer, two independent Frobenius series exist. If the roots are equal, the second solution involves \(\ln x\) times the first. If they differ by a positive integer, the second solution may or may not need a logarithm. These cases mirror the Cauchy-Euler patterns.
Bessel's equation
Bessel's equation \(x^2y'' + xy' + (x^2 - \nu^2)y = 0\) has a regular singular point at 0 with indicial roots \(\pm\nu\). Its solutions, Bessel functions \(J_\nu\) and \(Y_\nu\), describe vibrating drumheads, heat in cylinders, and electromagnetic waves in cables.
x^2y'' + xy' + (x^2 - \nu^2)y = 0Bessel functions in practice
\(J_0(x)\) looks like a decaying cosine: it oscillates with slowly shrinking amplitude. The zeros of \(J_0\) determine the frequencies at which a circular drum vibrates, which is why drums do not produce neat harmonic overtones like strings do.
Worked example
For \(2x^2y'' + xy' - y = 0\) (a Cauchy-Euler case), \(p_0 = \frac{1}{2}\), \(q_0 = -\frac{1}{2}\). The indicial equation \(r(r-1) + \frac{r}{2} - \frac{1}{2} = 0\) simplifies to \(2r^2 - r - 1 = 0\), with roots \(1\) and \(-\frac{1}{2}\). So solutions behave like \(x\) and \(x^{-1/2}\) near 0.
Common mistakes
Applying Frobenius at irregular singular points, where it fails. Forgetting \(a_0 \ne 0\). Overlooking the logarithmic case. Computing \(p_0\) and \(q_0\) without multiplying by \(x\) and \(x^2\).
Key takeaways
At regular singular points, try \(y = x^r\sum a_nx^n\). The indicial equation \(r(r-1) + p_0r + q_0 = 0\) gives the exponents; root differences decide whether logarithms appear. Bessel's equation is the key example, with solutions describing circular vibrations and waves.
Flashcard
xp \text{ and } x^2q \text{ analytic}
Check your understanding
Bessel functions most naturally describe: