With a differential equation, we can associate initial conditions or boundary conditions, auxiliary conditions on the unknown function and its derivatives. If these conditions are specified at a single value of the independent variable, they are referred to as initial conditions and the combination of the differential equation and an appropriate number of initial conditions is called an initial value problem (IVP). If these conditions are specified at more than one value of the independent variable, they are referred to as boundary conditions and the combination of the differential equation and the boundary conditions is called a boundary value problem (BVP).
a) The logistic equation,
with initial condition ; for the solution is
b) The mass-spring system equation,
with the initial conditions , ; for m = 10, k = 140, a = 90, , , , the solution is
a) The differential equation,
with the boundary conditions , ; the solution .
b) The differential equation,
with boundary conditions ; the solution is , for c an arbitrary constant.