SOME QUADRATURE FORMULAE FOR LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

Abstract

For the general linear differential equation of the II order with analytical coefficients \(a(x)\) and \(b(x)\)

\[
y''+a(x)y'+b(x)y=0
\]

we can prove the possibility of integration by quadratures, in the form of a series of integrals and the general solution is given by (13). So, the Liouville theory for general solution, which is based on the recognition of the fundamental system of particular solutions, can be substituted by quadrature formulae:

\[
y=c_1y_1+c_2y_2=F(a(x),b(x)).
\]

It means that the recognition of a fundamental system of particular solutions \(\{y_1,y_2\}\) at first is not necessary. We will show that such a fundamental system can be always formed, without trials and guesses. It means that every equation (1) can be solved by quadratures without exclusion and we can reject the traditional claim that in the general case equation (1) cannot be solved by quadratures.

The "solved by quadratures" means not only the finite integrals of coefficients, but also the series of an infinite number of integrals of coefficients.