A note on non-separable solutions of linear partial differential equations

A method has been presented for constructing non-separable solutions of homogeneous linear partial differential equations of the type F(D, D′)W = 0, where $\text{D =}\text{?}\text{?x}\text{, D′ =}\text{?}\text{?y}$,

$\text{F(D,D′)=}\text{∑}\text{n}\text{r+s=0}\text{C}{}_{\mathrm{rs}}\text{D}{}^{\mathrm{r}}\text{D′}{}^{\mathrm{s}}\text{,}$

where c_rs are constants and n stands for the order of the equation. The method has also been extended for equations of the form Φ(D, D′, D″)W = 0, where $\text{D =}\text{?}\text{?x}\text{, D′ =}\text{?}\text{?y}\text{, D″ =}\text{?}\text{?z}$ and

$\text{Φ(D,D′,D″)W=}\text{∑}\text{n}\text{r+s+t=0}\text{C}{}_{\mathrm{rst}}\text{D}{}^{\mathrm{r}}\text{D′}{}^{\mathrm{s}}\text{D}{}^{\mathrm{t}}\text{D″}{}^{\mathrm{s}}\text{.}$

As illustration, the method has been applied to obtain nonseparable solutions of the two and three dimensional Helmholtz equations.