Separable solution

A solution to a partial differential equation that factors into functions of separate variables, for example $u(x,t)=X(x)T(t)$. For the heat equation $u_t=D{u}_{xx}$, substitution yields $T^{\prime }/(DT)=X^{\prime \prime }/X$, so both sides are a common constant wherever division is valid. The resulting ordinary differential equations must be solved together with the original boundary conditions. Individual separated solutions generate more general solutions by superposition when the equation and boundary data are linear.