On approximate closed form solutions of linear ordinary differential equations

Approximate closed form solutions of a linear ordinary differential equation are obtained using piecewise approximations of the arbitrary coefficient function. We show how to obtain expressions for the general solution and the eigenvalue equation. An example is given with specific boundary conditions typical of a range of problems in mathematical physics and engineering. The method is robust and accurate and can be used as a complement to standard numerical techniques. The function representing the approximate solution has a simple form and can be used like a standard function in calculations that require the solution of the differential equation.