condition accompanying a differential equation in the solution of physical problems. In mathematical problems arising from physical situations, there are two considerations involved when finding a solution: (1) the solution and its derivatives must satisfy a differential equation, which describes how the quantity behaves within the region; and (2) the solution and its derivatives must satisfy other auxiliary conditions either describing the influence from outside the region (boundary values) or giving information about the solution at a specified time (initial values), representing a compressed history of the system as it affects its future behaviour. A simple example of a boundary-value problem may be demonstrated by the assumption that a function satisfies the equation f'(x)=2x for any x between 0 and 1 and that it is known that the function has the boundary value of 2 when x=1. The function f(x)=x2 satisfies the differential equation but not the boundary condition. The function f(x)=x2+1, on the other hand, satisfies both the differential equation and the boundary condition. The solutions of differential equations involve unspecified constants, or functions in the case of several variables, which are determined by the auxiliary conditions
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