A method for computation of integrals of solutions to differential equations (Q1905692)

A method is proposed for computing infinite integrals of the form \(J_{2k}= \int^\infty_0 \xi^{2k} (t) dt\), for \(k= 2, 3, \dots\) where \(\xi(t)\) is an asymptotically stable solution to the equation: \(\sum^n_{i=0} a_i {{d^i \xi (t)} \over {dt^i}} =0\), \(a_n =1\) with given initial conditions: \[ \xi^{(i)} (0)= {{d^i \xi(t)} \over {dt^i}} \biggl|_{t=0} \qquad \text{for } i= 0, 1, \dots, n-1, \] and \(a_i\) are given real numbers.