open interval I gives rise to the succession of derivatives 9 dX d x d X dt dt dt Now a differential equation in the function X t is therefore an equation of the type X MATHEMATICS F1 X du x X dX d X dt d 2 du x x 1 i n Such an equation is said to be an ordinary differential equation Thus an ordinary differential equation is a differential equation in which the constituent function X X t is a function of a single real variable Tim We often use the acronym ODE in place of the full term ordinary differential equation j1 i j n D u de X dik II On the other hand there are differential equations in a function u xu x of a real multivariable x x x x which ranges in an open subset of R Such a function u u x gives rise to mixed partial derivatives 0 Fx u 1 GJEJ 65 for various multi indices a a a with o Zi 0 1 2 the mixed partial derivative D u having the order a o 0 Now a differential equation in such a function u ux of a multi variable x x ranging in an open subset of R is an equation of the type du d u Du o m 0 2 x x dx its order being m Equation 2 is said to be a partial differential equation in u x because it involves the mixed partial derivatives of u We use the acronym PDE for this type of differential equations There is more about the setting of a differential equation In a mathematical problem a differential equation is accompanied by auxiliary data A solution of a differential equation is required to satisfy this auxiliary data To be more specific we are given a subset of the domain of a prospective solution and some of its derivatives of the solution at the points of this subset