= Sine-mode expansion for a quadratically degenerate elliptic equation
{title2=$b_n(y)=A_ny^{(1+\sqrt{1+4n^2})/2}$}
For $u_{xx}+y^2u_{yy}=0$ with zero vertical-side values on $0<x<\pi$, the <Fourier coefficient> $b_n(y)$ obeys the <Cauchy-Euler differential equation> $y^2b_n^{\prime\prime}=n^2b_n$. Boundedness at $y=0$ removes the negative <indicial root>, leaving $b_n(y)=A_ny^{(1+\sqrt{1+4n^2})/2}$. Bounded top coefficients give absolute convergence of the <Fourier sine series> below the top boundary.
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