We use non-polynomial spline with variable mesh to establish a numerical scheme for the solution of boundary value problem with singularity. The discrete equation of the problem is developed based on the condition of the class C1 of non-polynomial spline at the inner nodes and it is not valid for singularity. At singularity t = 0, the problem is modified in order to have a three term relationship. The method’s tridiagonal scheme is analyzed using the well-known discrete imbedding invariant algorithm. We discuss the error analysis of the scheme and two examples with layer at one end of the boundary are consider to demonstrate the practical utility of the scheme. Maximum absolute errors are present in tabular form to show the efficiency of the proposed method.