initial value problem

An initial value problem is then a differential equation (ordinary or partial, or even a system) which, besides of stating the relation among the derivatives, also, by giving the initial conditions, specifies the values of the unknowns at some point. This makes it possible to get a unique solution from the infinite number of potential ones. However, it should be noted that not all possible choices of initial conditions specify a unique solution. It is also possible that there will be more than one solution satisfying a particular choice of initial conditions, or perhaps no solution. In fact, one of the main problems in the theory of differential equations is to determine what sort of initial conditions will single out a unique solution of a given differential equation.