In Rani et al. (Numerical inversion of Laplace transform based on Bernsteinoperational matrix, Mathematical Methods in the Applied Sciences (2018) pp. 1–13), a numerical method is developed to find the inverse Laplace transform of certain functions using Bernstein operational matrix. Here, we describe Bernstein operational matrix of integration and propose an algorithm to solve linear time-varying systems governing differential equations. Apart from discussing error estimate, the method is implemented to linear differential equations on Bessel equation of order zero, damped harmonic oscillator, some higher order differential equations, singular integral equation, Volterra integral and integro-differential equations and nonlinear Volterra integral equations of the first kind. A comparison with some existing methods like Haar operational matrix, block pulse operational matrix and others are discussed. The method is simple and easy to implement on a variety of problems. Relative errors estimate just for 5th or 6th approximation show high applicability of the method.