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Nonhomogeneous matrix equations of the form Ax=b (1) can be solved by taking the matrix inverse to obtain x=A^(-1)b. (2) This equation will have a nontrivial solution iff the ...

A second-order ordinary differential equation of the form

The system of partial differential equations u_t+u_x = v^2-u^2 (1) v_t-v_x = u^2-v^2. (2)

The partial differential equation u_(xx)+(y^2)/(1-(y^2)/(c^2))u_(yy)+yu_y=0.

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 partial/(partialx)(u_t+uu_x+1/2u_(xxx)+u/(2t))+(3alpha^2)/(2t^2)u_(yy)=0 which arises in the study of water waves.

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.