Different types of problems in physics, for example, correspond different types of partial differential equations. The methods how to solve these equations differ from type to type.
The classification of differential equations follows from one single question: can we calculate formally the solution if sufficiently many initial data are given? Consider the initial problem for an ordinary differential equation \(y'(x)=f(x,y(x))\), \(y(x_0)=y_0\). Then one can determine formally the solution, provided the function \(f(x,y)\) is sufficiently regular. The solution of the initial value problem is formally given by a power series. This formal solution is a solution of the problem if \(f(x,y)\) is real analytic according to a theorem of Cauchy. In the case of partial differential equations the related theorem is the Theorem of Cauchy-Kowalevskaya. Even in the case of ordinary differential equations the situation is more complicated if \(y'\) is implicitly defined, i. e., the differential equation is \(F(x,y(x),y'(x))=0\) for a given function \(F\).
Thumbnail: Photograph of the Russian Mathematician Sofja Wassiljewna Kowalewskaja, the first major Russian female mathematician and responsible for important original contributions to analysis, partial differential equations and mechanics.