This paper presents a comparative study between neural network-based approaches and traditional numerical methods for solving differential equations and eigenvalue prob lems. We first applied a neural network to solve the one dimensional diffusion equation and compared the results with those obtained using the forward Euler iteration scheme. While the neural network provided flexibility in grid selection and was not constrained by stability criteria, it was less accurate and slower than the iterative method. Subsequently, we employed a neural network to determine the largest and smallest eigenvalues of a symmetric, real matrix. Though the neural network success fully converged to the correct eigenvalues and eigenvectors, it exhibited challenges in convergence time and accuracy, particu larly when compared to standard library routines. The findings demonstrate that while neural networks can replicate the results of traditional methods, they fall short in terms of computational efficiency and precision. Therefore, their use in these applications is limited by the trade-off between flexibility and performance.