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Orthogonality to a one parameter family of eigenfunctions
Let $\rho>0$ be a smooth realvalued function such that $\rho=1$ outside the unit interval $(-1,1)$. For each $t>0$, let us denote by $\{\lambda_n(t)\}_{n=1}^{\infty}$ and $\{\phi_n(t;x)\}_{n=1}^{...
- 3,987
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votes
1
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Legendre differential equation with additional term
In an application I encountered the ODE
$$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f
\left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x
\right) \right) \left( 8\,...
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