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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.

The partial differential equation u_(yy)=yu_(xx).

The partial differential equation del ^2A=-del xE, where del ^2 is the vector Laplacian.

An ordinary differential equation (frequently called an "ODE," "diff eq," or "diffy Q") is an equality involving a function and its derivatives. An ODE of order n is an ...

(dy)/(dx)+p(x)y=q(x)y^n. (1) Let v=y^(1-n) for n!=1. Then (dv)/(dx)=(1-n)y^(-n)(dy)/(dx). (2) Rewriting (1) gives y^(-n)(dy)/(dx) = q(x)-p(x)y^(1-n) (3) = q(x)-vp(x). (4) ...