On necessary conditions for resonance in turning point problems for ordinary differential equations (Q1179304)

The author investigates formal asymptotic expansion of the singularly perturbed turning point problem \(\varepsilon y''+f(x,\varepsilon)y'+g(x,\varepsilon)y=0\), \(y(-a)=\alpha\), \(y(- b)=\beta\), in the resonance case, i.e. when \(f(0,0)=0\), \(f_ x(0,0)<0\). He shows that each coefficient \(y_ k(x)\) of the outer expansion \(Y_ \varepsilon(x)=\sum^ \infty_{i=0}y_ i(x)\varepsilon^ i\) satisfies an equation whose solvability implies conditions on the coefficients of the equation. This leads to a sequence of necessary conditions for an outer expansion to exist which generalises classical conditions such as \(-g(0,0)/f_ x(0,0)\in\mathbb{N}\). Several examples are worked out for which explicit necessary conditions are obtained.