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The partial differential equation R[u](u_(rr)+(u_r)/r+u_(zz))=u_r^2+u_z^2, where R[u] is the real part of u (Calogero and Degasperis 1982, p. 62; Zwillinger 1997, p. 131).

The partial differential equation u_t=Du_(xx)+u-u^2.

The partial differential equation

The partial differential equation P_t=P_(xx)-uP_x+partial/(partialx){[u-F(x)]P}.

The partial differential equation u_t=u_(xxxxx)+5/2u_(xxx)u+(25)/4u_(xx)u_x+5/4u^2u_x.

The partial differential equation (Cole and Cook 1986, p. 34; Zwillinger 1997, p. 134).

The partial differential equation ((partial^2)/(partialt^2)-(partial^2)/(partialx^2))((u_(xy))/u)+2(u^2)_(xt)=0.

The wave equation is the important partial differential equation del ^2psi=1/(v^2)(partial^2psi)/(partialt^2) (1) that describes propagation of waves with speed v. The form ...

A solution of a linear homogeneous ordinary differential equation with polynomial coefficients.

The partial differential equation u_(xt)=sinhu, which contains u_(xt) instead of u_(xx)-u_(tt) and sinhu instead to sinu, as in the sine-Gordon equation (Grauel 1985; ...