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The Helmholtz differential equation is not separable in toroidal coordinates

The third-order ordinary differential equation y^(''')+alphayy^('')+beta(1-y^('2))=0.

The Helmholtz differential equation is not separable in bispherical coordinates.

Whittaker and Watson (1990, pp. 539-540) write Lamé's differential equation for ellipsoidal harmonics of the first kind of the four types as ...

The ordinary differential equation y^('')+k/xy^'+deltae^y=0.

Bessel's correction is the factor (N-1)/N in the relationship between the variance sigma and the expectation values of the sample variance, <s^2>=(N-1)/Nsigma^2, (1) where ...

For x>0, J_0(x) = 2/piint_0^inftysin(xcosht)dt (1) Y_0(x) = -2/piint_0^inftycos(xcosht)dt, (2) where J_0(x) is a zeroth order Bessel function of the first kind and Y_0(x) is ...

In bipolar coordinates, the Helmholtz differential equation is not separable, but Laplace's equation is.

The second-order ordinary differential equation y^('')-[(m(m+1)+1/4-(m+1/2)cosx)/(sin^2x)+(lambda+1/2)]y=0.

The second-order ordinary differential equation y^('')+[alpha/(cosh^2(ax))+betatanh(ax)+gamma]y=0.