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).
Best Answer
This is called the Taylor series method. An older similar question and answer with further links is Higher-order corrections for Euler's method
The expansion of the solution in terms of $f$ and its derivatives is the departure point to construct numerical methods and derive their order conditions. This is formalized in B-series over rooted trees and their Hopf algebra, "B" standing for their inventor Butcher, who is also responsible for the Butcher tableaux.
Indeed the recursive formulation of the derivatives is useful in implementing this method, one example is TADIFF (Taylor (series) arithmetic) in the FADBAD (forward/backward algorithmic differentiation) package.