• ### The Derivative of a Solution to a Second Order Parameter Dependent Boundary Value Problem with a Nonlocal Integral Boundary Condition

Jeffrey W. Lyons and Joseph K. Miller

Abstract: We discuss derivatives of the solution of the second order parameter dependent boundary value problem with an integral boundary condition$y” = f(x,y,y’,\lambda ), y(x_{1})=y_{1}, y(x_2 ) + \int_c^d \,ry(x)dx = y_2$and its relationship to a second order nonhomogeneous differential equation which corresponds to the traditional variational equation. Specifically, we show that given a solution y(x) of the boundary value problem, the derivative of the solution with respect to the parameter λ is itself a solution to the aforementioned nonhomogeneous equation with interesting boundary conditions.