Numerical solution of first-order exact differential equations by the integrating factor method

A numerical algorithm for solving exact differential equations is proposed, based both on the efficient calculation of integrating factors and on a “new” numerical method for integrating functions. Robust determination of the integrating factors is implemented by using the Chebyshev interpolation of the desired functions and performing calculations on Gauss – Lobatto grids, which ensure the discrete orthogonality of the Chebyshev matrices. After that, the integration procedure is carried out using the Chebyshev integration matrices. The integrating factor and the final potential of the ODE solution are presented as interpolation polynomials depending on a limited number of numerically recoverable expansion coefficients.