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The partial differential equation u_t+2uu_x-nuu_(xx)+muu_(xxx)=0.

In two-dimensional bipolar coordinates, Laplace's equation is ((coshv-cosu)^2)/(a^2)((partialF^2)/(partialu^2)+(partialF^2)/(partialv^2))=0, which simplifies to ...

The second-order ordinary differential equation y^('')+alpha(x)y^'+x^2y^n=0.

In parabolic cylindrical coordinates, the scale factors are h_u=h_v=sqrt(u^2+v^2), h_z=1 and the separation functions are f_1(u)=f_2(v)=f_3(z)=1, giving Stäckel determinant ...

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

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

The Helmholtz differential equation is not separable in bispherical coordinates.

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

The partial differential equation u_t+u_x-6uu_x-u_(txx)=0.

The second-order ordinary differential equation xy^('')+(c-x)y^'-ay=0, sometimes also called Kummer's differential equation (Slater 1960, p. 2; Zwillinger 1997, p. 124). It ...