The solution of a linear two-point boundary value problem with time-varying coefficients is obtained by Fourier series approximation. Properties of Fourier series are first briefly discussed and the operational matrix of integration is presented. A transformation matrix relating the back vector to the current time vector together with the operational matrix are utilized to reduce the system to a set of linear algebraic equations and thereby obtain the approximate solution. Illustrative examples are included to demonstrate the validity and applicability of the technique.