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The ordinary differential equation

A partial differential equation of second-order, i.e., one of the form Au_(xx)+2Bu_(xy)+Cu_(yy)+Du_x+Eu_y+F=0, (1) is called parabolic if the matrix Z=[A B; B C] (2) ...

The partial differential equation u_t=del ·(u^mdel u).

The partial differential equation u_(tt)-u_(xx)=epsilon(u_t-u_t^3).

The method for solving the Goursat problem and Cauchy problem for linear hyperbolic partial differential equations using a Riemann function.

The partial differential equation u_t+45u^2u_x+15u_xu_(xx)+15uu_(xxx)+u_(xxxxx)=0.

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; ...

The partial differential equation del ^2u+lambda^2sinhu=0, where del ^2 is the Laplacian (Ting et al. 1987; Zwillinger 1997, p. 135).

The partial differential equation u_(xy)+alphau_x+betau_y+gammau_xu_y=0.

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