Derivatives & Differential Equations
For the differential equations applicable to physical problems, it is often possible to start with a general form and force that form to fit the physical boundary conditions of the problem. This kind of approach is made possible by the fact that there is one and only one solution to the differential equation, i.e., the solution is unique.
Stated in terms of a first order differential equation, if the problem
meets the condition such that f(x,y) and the derivative of y is continuous in a given rectangle of (x,y) values, then there is one and only one solution to the equation which will meet the boundary conditions.