A numerical method for solving the Cauchy problem for ODEs using a system of polynomials generated by a system of modified Laguerre polynomials

In this paper, we consider a numerical realization of an iterative method for solving the Cauchy problem for ordinary differential equations, based on representing the solution in the form of a Fourier series by the system of polynomials $\{L_{1,n}(x;b)\}_{n=0}^\infty$, orthonormal with respect to the Sobolev-type inner product
$$ \langle f,g\rangle=f(0)g(0)+\int_{0}^\infty f'(x)g'(x)\rho(x;b)dx $$
and generated by the system of modified Laguerre polynomials $\{L_{n}(x;b)\}_{n=0}^\infty$, where $b>0$. In the approximate calculation of the Fourier coefficients of the desired solution, the Gauss–Laguerre quadrature formula is used.