On bounds for monotonic first order differential systems and the Liouville–Green approximation

Many special functions are solutions of first order linear systems y′(x)=a(x)y(x)+d(x)w(x)y′(x)=a(x)y(x)+d(x)w(x), w′(x)=b(x)w(x)+e(x)y(x)w′(x)=b(x)w(x)+e(x)y(x). We obtain bounds for the logarithmic derivatives of the solutions of monotonic systems satisfying certain initial conditions. In particular, by Liouville-transforming the first order system equivalent to the second order ODE y″(x)+A(x)y(x)=0y″(x)+A(x)y(x)=0 we obtain bounds related to the Liouville–Green approximation and find conditions under which such approximation is a bound for some solutions. We illustrate this with the Airy equation.