| We consider the linear differential equation $\epsilon y''+\varphi(x,\epsilon)y'+\psi(x,\epsilon)y=0$ where $\epsilon>0$ is a small parameter and where $x,\varphi(x,0)>0$ if $x\neq0$. We suppose that $\varphi$ and $\psi$ are real analytic on $[a,b]\times{0}$ and that the function $\psi_0:x\mapsto\psi(x,0)$ has a zero at $x=0$ of at least the same order as $\varphi_0:x\mapsto\varphi(x,0)$. We call {\em resonant solution} a (family of) solution $y_\epsilon:[a,b]\rightarrow{\mathbf R}$ tending uniformly to a non trivial solution of the reduced equation $\varphi(x,0)y'+\psi(x,0)y=0$ obtained by formally replacing $\epsilon$ by 0 and such that all its derivatives remain bounded as $\epsilon$ tends to 0. We prove that the existence of a formal series solution whose coefficients have no poles at $x=0$ is a necessary and sufficient condition for the equation to have a resonant solution. This generalizes a result of C.H. Lin. Our proof is based on the study of overstable solutions of a corresponding Riccati equation. The main tool is a "principle of analytic continuation'' for overstable solutions of analytic ordinary differential equations of first order, which is also of independent interest. |
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