Initial Conditions and Boundary Conditions
In many problems on integration, an initial condition (y = y_{0} when x = 0) or a boundary condition (y = y_{0} when x = x_{0} ) is given which uniquely determines the constant of integration. As a result of the unique determination of 'c', a specific curve can be singled out from a family of curves.
Example
For the boundary condition, y = 15 when x = 2, the integral y = ∫4dx is evaluated as follows:
y = ∫4dx = 4x + c
Substituting, y = 15, when x = 2
15 = 4(2) + c or c = 7
y = 4x + 7
Note that even though 'c' is specified, ∫4dx remains an indefinite integral because x_{n} is unspecified. Thus, the integral 4x + 7 can assume an infinite number of possible values.