Let $y^{\prime}=f\left( x,y\right) $ be first order ode with expansion around $x_{0}$ with initial conditions $y\left( x_{0}\right) =y_{0}$, and where $f\left( x,y\right) $ is analytic at $x_{0}$ then the solution to the ode in series expansion around $x_0$ by Taylor series is given by
$$ y=y_{0}+\sum_{n=1}^{\infty}\frac{x^{n}}{n!}\left. F_{n}\left( x,y\right) \right\vert _{x=x_{0},y=y_{0}} $$
Where
\begin{align*} F_{1}\left( x,y\right) & =f\left( x,y\right) \\ F_{n+1}\left( x,y\right) & =\frac{\partial F_{n}}{\partial x}+\left( \frac{\partial F_{n}}{\partial y}\right) F_{1} \end{align*}
What would be the equivalent formulation (if one exists) for a second order ode $y^{\prime\prime}=f\left( x,y,y^{\prime}\right) $ with initial conditions $y\left( x_{0}\right) =y_{0},y^{\prime}\left( x_{0}\right) =y_{0}^{\prime}$ where it is assumed also that $f\left( x,y,y^{\prime}\right) $ is analytic at $x_{0}$?
The above was taken from sympy ode solver here with reference for the above formula given as
Travis W. Walker, Analytic power series technique for solving first-order differential equations, p.p 17, 18
But I could not find such book searching. I could only find this page which references paper
Walker, T.W. “Analytic Power Series Technique for Solving First-Order Ordinary Differential Equations.” MAA Rocky Mountain Section 2008 Meeting, Spearfish, South Dakota. 25-26 Apr. 2008.
Where the above formulation is given. But all links from the above page are dead now.
One advantage of this formulation over standard power series, is that is it easier to automate and to program. (no need to find recurrence relation for the $a_n$ for example). The $F_n$ expressions above do this job already.
Is it possible to extend the above for second order ode by use of Taylor series expansion? We would need additional term for the partial derivative of $f(x,y,y')$ w.r.t. $y'$ ofcourse.
Does any one knows of a reference where it is given in the above form? (compared to the standard power series form, for ordinary point).
