Initial Conditions and Boundary Conditions

In many problems on integration, an initial condition (y = y₀ when x = 0) or a boundary condition (y = y₀ when x = x₀ ) 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.