Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2006 / Paper 1 / 6

For the normalized second-order equation $y^{\prime \prime }+P(x)y^{\prime }+Q(x)y=0$, an ordinary point has analytic $P,Q$. A regular singular point is a singular point where $(x-x_0)P$ and $(x-x_0{)}^{2}Q$ extend analytically through $x_0$. The two finite singular points here are $0$ and $1/2$. At zero, the respective limiting coefficients are $1/2,0$; at $1/2$, they are $-1,0$. Thus both are regular singular points, with indicial roots $0,1/2$ and $0,2$, respectively.

Use the Frobenius method and write $y=x^r{\sum }_{n\ge 0}{a}_n{x}^n$, $a_0\ne 0$, with coefficients at negative indices defined as zero. The lowest power gives the indicial equation $r(2r-1)=0$, so $r=0$ or $r=1/2$. Equating the coefficient of $x^{n+r-1}$ gives, for $n\ge 1$,For $r=0$ and $a_0=1$, the first coefficients are $1,2,2,4/3,\dots$. For $r=1/2$ they are $1,1,1/2,1/6,\dots$, multiplying $x^{1/2}$. The distinct nonintegrally separated indicial roots give two independent Frobenius series. The half-power solution is a generalized power series, rather than an ordinary Taylor series at zero.

The coefficients in the Taylor series areFor either series, the ratio test gives absolute term ratio $|x|/(n+1)$ or $2|x|/(n+1)$, tending to zero for each fixed real or complex $x$. Both have infinite radius of convergence.

Substituting $a_n=2^n/n!$ into the $r=0$ recurrence, and $a_n=1/n!$ into the $r=1/2$ recurrence, satisfies every coefficient equation. They give $e^{2x}$ and $\sqrt{x}{e}^x$. Thus on a positive interval not crossing a singular point,Their Wronskian is $e^{3x}(2x-1)/(2\sqrt{x})$, nonzero for $x>0$, $x\ne 1/2$, proving independence on each ordinary interval. Both functions are analytic near $1/2$, so this is an apparent regular singular point: a pole in normalized coefficients need not force a singular solution. The Wronskian can vanish at that singular point without making the two functions linearly dependent on an ordinary interval.