Series Solutions of Differential Equations

Definition To solve a linear differential equation near a point $x_0$, assume a solution of the form

with radius of convergence $\rho > 0$. On $\lvert x - x_0 \rvert < \rho$ the series may be differentiated term by term, with the same radius,

Substitute these into the equation and determine the $a_n$. The values $a_0 = y(x_0)$ and $a_1 = y'(x_0)$ carry the initial data.

Definition After substitution, shift the summation index in each sum so that every sum is written with the same power $(x - x_0)^n$. Since a power series is zero on an interval only if all its coefficients are zero, the coefficient of each power must vanish. The resulting formula that gives later coefficients from earlier ones is the recurrence relation.

Definition Write the equation $P(x)y'' + Q(x)y' + R(x)y = 0$ in the form $y'' + p(x)y' + q(x)y = 0$ with $p = Q/P$ and $q = R/P$. A point $x_0$ is an ordinary point if $p$ and $q$ are analytic at $x_0$, meaning each equals a power series in $x - x_0$ with positive radius. Otherwise $x_0$ is a singular point. A singular point is regular if the functions

are both analytic at $x_0$, and irregular otherwise. When $P$, $Q$, $R$ are polynomials with no common factor, $x_0$ is ordinary exactly when $P(x_0) \ne 0$.

Definition Theorem. If $x_0$ is an ordinary point and the series of $p$ and $q$ about $x_0$ both converge for $\lvert x - x_0 \rvert < \rho$, then every solution of $y'' + p\,y' + q\,y = 0$ is a power series about $x_0$ that converges at least for $\lvert x - x_0 \rvert < \rho$. In particular, for polynomial coefficients with no common factor, the guaranteed radius is at least the distance from $x_0$ to the nearest zero of $P$ in the complex plane.

Definition Let $0$ be a regular singular point, and suppose $x\,p(x) = \sum p_n x^n$ and $x^2 q(x) = \sum q_n x^n$ converge for $\lvert x \rvert < \rho$. Try

The lowest power gives the indicial equation

Theorem. Assume $p$ and $q$ are real, so the indicial roots are real or a complex conjugate pair, and take $x$ in $0 < x < \rho$. In each case the series below converge for $\lvert x \rvert < \rho$. When the roots are real, label them $r_1 \ge r_2$.

1. For real roots there is always a solution $y_1 = x^{r_1}\sum a_n x^n$ with $a_0 = 1$.
2. If $r_1 - r_2$ is not an integer, a second, linearly independent solution is $y_2 = x^{r_2}\sum b_n x^n$ with $b_0 = 1$.
3. If $r_1 = r_2$, a second solution is $y_2 = y_1 \ln x + x^{r_1}\sum_{n \ge 1} b_n x^n$, so a logarithm always appears.
4. If $r_1 - r_2$ is a positive integer, a second solution is $y_2 = a\,y_1 \ln x + x^{r_2}\sum b_n x^n$ with $b_0 = 1$, where the constant $a$ may or may not be zero.
5. Complex roots $\lambda \pm i\mu$ with $\mu \ne 0$. The method works with complex coefficients, and the real and imaginary parts of the solution for $r = \lambda + i\mu$ are two independent real solutions. Near $0$ they behave like $x^{\lambda}\cos(\mu \ln x)$ and $x^{\lambda}\sin(\mu \ln x)$, since $x^{i\mu} = e^{i\mu \ln x}$.

For $-\rho < x < 0$ the same statements hold with $x^r$ and $\ln x$ replaced by $\lvert x \rvert^r$ and $\ln \lvert x \rvert$.

Definition For a constant $\nu \ge 0$ the Bessel equation of order $\nu$ is

The point $0$ is a regular singular point with $p_0 = 1$ and $q_0 = -\nu^2$, so the indicial roots are $\pm\nu$. The root $r = \nu$ gives the Bessel function of the first kind,

which converges for every $x > 0$, and for every real $x$ when $\nu$ is an integer. The indicial roots differ by $2\nu$. If $\nu$ is not an integer, $J_{-\nu}$, given by the same series with $-\nu$ in place of $\nu$, is a second solution independent of $J_\nu$, even when $2\nu$ is an odd integer. If $\nu = n$ is an integer, then $J_{-n} = (-1)^n J_n$ is not new, and case 3 or 4 of the Frobenius theorem gives a second solution with a term $J_n(x)\ln x$. The standard choice of second solution for every $\nu \ge 0$ is the Bessel function of the second kind $Y_\nu$, so the general solution on $x > 0$ is

The function $Y_\nu$ is unbounded as $x \to 0^+$, which is why a drum, whose center is at $x = 0$, uses only $J_\nu$. Functions defined this way, by a series or an integral rather than a finite formula, are called special functions.