An approximate solution of the Cauchy problem for an ODE system by means of system $1,\, x,\, \{\frac{\sqrt{2}}{\pi n}\sin(\pi nx)\}_{n=1}^\infty$

We consider a system of functions $\xi_0(x)=1,\, \{\xi_n(x)=\sqrt{2}\cos(\pi nx)\}_{n=1}^\infty$ and the system
$$ \xi_{1,0}(x)=1,\, \xi_{1,1}(x)=x,\, \xi_{1,n+1}(x)=\int_0^x \xi_{n}(t)dt=\frac{\sqrt{2}}{\pi n}\sin(\pi nx),\, n=1,2,\ldots, $$
generated by it, which is Sobolev orthonormal with respect to a scalar product of the form $<f,g>=f'(0)g'(0)+\int_{0}^{1}f'(t)g'(t)dt$. It is shown that the Fourier series and sums with respect to the system $\{\xi_{1,n}(x)\}_{n=0}^\infty$ are a convenient and very effective tool for the approximate solution of the Cauchy problem for systems of nonlinear ordinary differential equations (ODEs).