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A ghost perturbation scheme to solve ordinary differential equations

We propose an algebraic method that finds a sequence of functions that exponentially approach the solution of any second-order ordinary differential equation (ODE) with any boundary conditions. We define an extended ODE (eODE) composed of a linear generic differential operator that depends on free parameters, $p$, plus an $ε$ perturbation formed by the original ODE minus the same linear term. After the eODE&#39;s formal $ε$ expansion of the solution, we can solve order by order a hierarchy of linear ODEs, and we get a sequence of functions $y_n(x;ε,p)$ where $n$ indicates the number of terms that we keep in the $ε$-expansion. We fix the parameters to the optimal values $p^*(n)$ by minimizing a distance function of $y_n$ to the ODE&#39;s solution, $y$, over a given $x$-interval. We see that the eODE&#39;s perturbative solution converges exponentially fast in $n$ to the ODE solution when $ε=1$: $\vert y_n(x;ε=1,p^*(n))-y(x)\vert<Cδ^{n+1}$ with $δ<1$. The method permits knowing the number of solutions for Boundary Value Problems just by looking at the number of minima of the distance function at each order in $n$, $p^{*,α}(n)$, where each $α$ defines a sequence of functions $y_n$ that converges to one of the ODE&#39;s solutions. We present the method by its application to several cases where we discuss its properties, benefits and shortcomings, and some practical algorithmic improvements.