There are classical techniques based on differential and integral calculus to solve differential equations analytically. However, as the number of variables and/or the order of differential equation increase, finding a closed form solution using analytical methods becomes very tedious and sometimes almost impossible. This difficulty is further compounded by the complex geometry of the domain over which the solution has to be obtained. Hence, numerical approximation is a must and many numerical methods are developed so as to approximate the solution of differential equation(s). Finite difference method (FDM), Finite element method (FEM), Finite volume method (FVM) etc. are the different numerical techniques used widely in engineering and sciences. However, these methods generate a discrete solution and require mesh generation, which is highly expensive and needs specialized algorithms. In contrast the RBF-NN (Radial Basis Function Neural Network) method attracted intensive research in recent years for it is a true meshless method thus avoids the cost of mesh generation and generates a closed form solution. The RBF-NN method involves in satisfying the differential equation in the domain and the constraints at the boundaries. Certain points in the domain and on the boundaries are taken as training data and are called collocation points, at which the network is trained. In the present work certain problems in 1D and heat conduction problems in 2D are solved using RBF-NN to check its effectiveness. The method is applied to 2D torsion problem for non-circular shafts in detail. The results indicate great potential of the method to be an alternative to the other mesh dependant methods.