7A (source code)
= 7A
{scope}
{title2=Differential Equations}
\b[Classification and characteristic exponents.] For a normalized <second-order linear differential equation>, an <ordinary point> at zero means $p$ and $q$ are <real analytic functions> there. A <regular singular point> is a singular point where $xp$ and $x^2q$ nevertheless extend as <real analytic functions> to zero. Write $p_0=\lim_{x\to0}xp(x)$ and $q_0=\lim_{x\to0}x^2q(x)$. The lowest-power coefficient in the <Frobenius method> gives the <indicial equation>
$$
\sigma(\sigma-1)+p_0\sigma+q_0=0.
$$
Its roots are the <characteristic exponents at a regular singular point>. If their difference is not an integer, there are two <linearly independent> solutions $x^{\sigma_j}A_j(x)$, with $A_j$ <real analytic> and $A_j(0)\ne0$. For distinct real roots $\sigma_1>\sigma_2$ with integer difference, a solution for the larger root exists, and a second has the form $x^{\sigma_2}B(x)+c\,y_1\log x$; the logarithmic coefficient can vanish. For a repeated root, the second solution has the form $y_1\log x+x^{\sigma_1}B(x)$ after normalization. This is a <Logarithmic Frobenius solution>. Complex conjugate roots $a\pm ib$ give real solutions whose leading behaviour is $x^a\cos(b\log x)$ and $x^a\sin(b\log x)$, with analytic-series corrections. All these descriptions are on $x>0$, or on a fixed complex branch.
For the present equation, division by $x$ gives $p=1+(1-m)/x$ and $q=(1-m)/x$. Therefore $xp=x+1-m$ and $x^2q=(1-m)x$ are <real analytic>. \b[Zero is regular singular when $m\ne1$; when $m=1$, it is ordinary after removing the common factor $x$.] In that exceptional case the normalized equation is $y''+y'=0$.
For the <Frobenius method> substitution, the lowest coefficient and the general <linear recurrence relation> are
$$
\boxed{\sigma(\sigma-m)=0,\qquad \sigma=0\ \text{or}\ m,}
$$
$$
(n+\sigma)(n+\sigma-m)a_n+(n+\sigma-m)a_{n-1}=0\quad(n\geq1).
$$
These equations come respectively from the coefficients of $x^{\sigma-1}$ and $x^{n+\sigma-1}$. At $m=1$, the same formal exponents $0,1$ describe the two possible leading powers at an <ordinary point>, rather than singular behaviour.
Back to article page
