ode internal name "higher_order_taylor_series_method_ordinary_point"
Only ordinary point is supported and for third order ode at this time using Taylor series (not power series) method. Let \[ y^{\prime \prime \prime }=f\left ( x,y,y^{\prime },y^{\prime \prime }\right ) \] Assuming expansion is at \(x_{0}=0\) (we can always shift the actual expansion point to \(0\) by change of variables) and assuming \(f\left ( x,y,y^{\prime },y^{\prime \prime }\right ) \) is analytic at \(x_{0}\) which must be the case for an ordinary point. Let initial conditions be \(y\left ( x_{0}\right ) =y_{0}\) and \(y^{\prime }\left ( x_{0}\right ) =y_{0}^{\prime }\) and \(y^{\prime \prime }\left ( x_{0}\right ) =y_{0}^{\prime \prime }\). Using Taylor series gives\begin {align*} y\left ( x\right ) & =y\left ( x_{0}\right ) +\left ( x-x_{0}\right ) y^{\prime }\left ( x_{0}\right ) +\frac {\left ( x-x_{0}\right ) ^{2}}{2}y^{\prime \prime }\left ( x_{0}\right ) +\frac {\left ( x-x_{0}\right ) ^{3}}{3!}y^{\prime \prime \prime }\left ( x_{0}\right ) +\frac {\left ( x-x_{0}\right ) ^{4}}{4!}y^{\prime \prime \prime \prime }\left ( x_{0}\right ) +\cdots \\ & =y_{0}+xy_{0}^{\prime }+\frac {x^{2}}{2}y_{0}^{\prime \prime }+\frac {x^{3}}{3!}\left . f\right \vert _{x_{0},y_{0},y_{0}^{\prime },y_{0}^{\prime \prime }}+\frac {x^{4}}{4!}\left . f^{\prime }\right \vert _{x_{0},y_{0},y_{0}^{\prime },y_{0}^{\prime \prime }}+\cdots \\ & =y_{0}+xy_{0}^{\prime }+\frac {x^{2}}{2}y_{0}^{\prime \prime }+\sum _{n=0}^{\infty }\frac {x^{n+3}}{\left ( n+3\right ) !}\left . \frac {d^{n}f}{dx^{n}}\right \vert _{x_{0},y_{0},y_{0}^{\prime },y_{0}^{\prime \prime }} \end {align*}
But \begin {align} \frac {df}{dx} & =\frac {\partial f}{\partial x}\frac {dx}{dx}+\frac {\partial f}{\partial y}\frac {dy}{dx}+\frac {\partial f}{\partial y^{\prime }}\frac {dy^{\prime }}{dx}+\frac {\partial f}{\partial y^{\prime \prime }}\frac {dy^{\prime \prime }}{dx}\tag {1}\\ & =\frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}y^{\prime }+\frac {\partial f}{\partial y^{\prime }}y^{\prime \prime }+\frac {\partial f}{\partial y^{\prime }}y^{\prime \prime \prime }\nonumber \\ & =\frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}y^{\prime }+\frac {\partial f}{\partial y^{\prime }}y^{\prime \prime }+\frac {\partial f}{\partial y^{\prime }}f\nonumber \\ \frac {d^{2}f}{dx^{2}} & =\frac {d}{dx}\left ( \frac {df}{dx}\right ) \nonumber \\ & =\frac {\partial }{\partial x}\left ( \frac {df}{dx}\right ) +\frac {\partial }{\partial y}\left ( \frac {df}{dx}\right ) y^{\prime }+\frac {\partial }{\partial y^{\prime }}\left ( \frac {df}{dx}\right ) y^{\prime \prime }+\frac {\partial }{\partial y^{\prime \prime }}\left ( \frac {df}{dx}\right ) y^{\prime \prime \prime }\tag {2}\\ & =\frac {\partial }{\partial x}\left ( \frac {df}{dx}\right ) +\frac {\partial }{\partial y}\left ( \frac {df}{dx}\right ) y^{\prime }+\frac {\partial }{\partial y^{\prime }}\left ( \frac {df}{dx}\right ) y^{\prime \prime }+\frac {\partial }{\partial y^{\prime \prime }}\left ( \frac {df}{dx}\right ) f\nonumber \\ \frac {d^{3}f}{dx^{3}} & =\frac {d}{dx}\left ( \frac {d^{2}f}{dx^{2}}\right ) \nonumber \\ & =\frac {\partial }{\partial x}\left ( \frac {d^{2}f}{dx^{2}}\right ) +\frac {\partial }{\partial y}\left ( \frac {d^{2}f}{dx^{2}}\right ) y^{\prime }+\frac {\partial }{\partial y^{\prime }}\left ( \frac {d^{2}f}{dx^{2}}\right ) y^{\prime \prime }+\frac {\partial }{\partial y^{\prime \prime }}\left ( \frac {d^{2}f}{dx^{2}}\right ) f\tag {3}\\ & \vdots \nonumber \end {align}
And so on. Hence if we name \(F_{0}=f\left ( x,y,y^{\prime },y^{\prime \prime }\right ) \) then the above can be written as \begin {align} F_{0} & =f\left ( x,y,y^{\prime }\right ) \tag {4}\\ F_{1} & =\frac {df}{dx}\nonumber \\ & =\frac {dF_{0}}{dx}\nonumber \\ & =\frac {\partial F_{0}}{\partial x}+\frac {\partial F_{0}}{\partial y}y^{\prime }+\frac {\partial F_{0}}{\partial y^{\prime }}y^{\prime \prime }+\frac {\partial F_{0}}{\partial y^{\prime \prime }}y^{\prime \prime \prime }\nonumber \\ & =\frac {\partial F_{0}}{\partial x}+\frac {\partial F_{0}}{\partial y}y^{\prime }+\frac {\partial F_{0}}{\partial y^{\prime }}y^{\prime \prime }+\frac {\partial F_{0}}{\partial y^{\prime \prime }}F_{0}\nonumber \\ F_{2} & =\frac {d}{dx}\left ( \frac {d}{dx}f\right ) \nonumber \\ & =\frac {d}{dx}\left ( F_{1}\right ) \nonumber \\ & =\frac {\partial F_{1}}{\partial x}+\frac {\partial F_{1}}{\partial y}y^{\prime }+\frac {\partial F_{1}}{\partial y^{\prime }}y^{\prime \prime }+\frac {\partial F_{1}}{\partial y^{\prime \prime }}y^{\prime \prime \prime }\nonumber \\ & =\frac {\partial F_{1}}{\partial x}+\frac {\partial F_{1}}{\partial y}y^{\prime }+\frac {\partial F_{1}}{\partial y^{\prime }}y^{\prime \prime }+\frac {\partial F_{1}}{\partial y^{\prime \prime }}F_{0}\nonumber \\ & \vdots \nonumber \\ F_{n} & =\frac {d}{dx}\left ( F_{n-1}\right ) \nonumber \\ & =\frac {\partial }{\partial x}F_{n-1}+\left ( \frac {\partial F_{n-1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{n-1}}{\partial y^{\prime }}\right ) y^{\prime \prime }+\left ( \frac {\partial F_{n-1}}{\partial y^{\prime \prime }}\right ) y^{\prime \prime \prime }\nonumber \\ & =\frac {\partial }{\partial x}F_{n-1}+\left ( \frac {\partial F_{n-1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{n-1}}{\partial y^{\prime }}\right ) y^{\prime \prime }+\left ( \frac {\partial F_{n-1}}{\partial y^{\prime \prime }}\right ) F_{0} \tag {6} \end {align}
Therefore (6) can be used from now on along with \begin {equation} y\left ( x\right ) =y_{0}+xy_{0}^{\prime }+\frac {x^{2}}{2}y_{0}^{\prime \prime }+\sum _{n=0}^{\infty }\frac {x^{n+3}}{\left ( n+3\right ) !}\left . F_{n}\right \vert _{x_{0},y_{0},y_{0}^{\prime },y_{0}^{\prime \prime }} \tag {7} \end {equation} To find \(y\left ( x\right ) \) series solution around \(x=0\).