problem 730

Internal problem ID [15473]
Internal ﬁle name [OUTPUT/15474_Wednesday_May_08_2024_04_01_09_PM_89370389/index.tex]

Book: A book of problems in ordinary diﬀerential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 18.1 Integration of diﬀerential equation in series. Power series. Exercises page 171
Problem number: 730.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "second order series method. Ordinary point", "second order series method. Taylor series method"

\[ \boxed {\ln \left (x \right ) y^{\prime \prime }-\sin \left (x \right ) y=0} \] With initial conditions \begin {align*} [y \left ({\mathrm e}\right ) = {\mathrm e}^{-1}, y^{\prime }\left ({\mathrm e}\right ) = 0] \end {align*}

23.7.1 Existence and uniqueness analysis

This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime \prime } + p(x)y^{\prime } + q(x) y &= F \end {align*}

Where here \begin {align*} p(x) &=0\\ q(x) &=-\frac {\sin \left (x \right )}{\ln \left (x \right )}\\ F &=0 \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }-\frac {\sin \left (x \right ) y}{\ln \left (x \right )} = 0 \end {align*}

The ode does not have its expansion point at \(x = 0\), therefore to simplify the computation of power series expansion, change of variable is made on the independent variable to shift the initial conditions and the expasion point back to zero. The new ode is then solved more easily since the expansion point is now at zero. The solution converted back to the original independent variable. Let \[ t = x -{\mathrm e} \] The ode is converted to be in terms of the new independent variable \(t\). This results in \[ \ln \left (t +{\mathrm e}\right ) \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )-\sin \left (t +{\mathrm e}\right ) y \left (t \right ) = 0 \] With its expansion point and initial conditions now at \(t = 0\). With initial conditions now becoming \begin {align*} y(0) &= {\mathrm e}^{-1}\\ y'(0) &= 0 \end {align*}

The transformed ODE is now solved. Solving ode using Taylor series method. This gives review on how the Taylor series method works for solving second order ode.

Let \[ y^{\prime \prime }=f\left ( x,y,y^{\prime }\right ) \] Assuming expansion is at \(x_{0}=0\) (we can always shift the actual expansion point to \(0\) by change of variables) and assuming \(f\left ( x,y,y^{\prime }\right ) \) 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 \(y^{\prime }\left ( x_{0}\right ) =y_{0}^{\prime }\). 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 ) +\cdots \\ & =y_{0}+xy_{0}^{\prime }+\frac {x^{2}}{2}\left . f\right \vert _{x_{0},y_{0},y_{0}^{\prime }}+\frac {x^{3}}{3!}\left . f^{\prime }\right \vert _{x_{0},y_{0},y_{0}^{\prime }}+\cdots \\ & =y_{0}+xy_{0}^{\prime }+\sum _{n=0}^{\infty }\frac {x^{n+2}}{\left ( n+2\right ) !}\left . \frac {d^{n}f}{dx^{n}}\right \vert _{x_{0},y_{0},y_{0}^{\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}\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 x}+\frac {\partial f}{\partial y}y^{\prime }+\frac {\partial f}{\partial y^{\prime }}f\\ \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 ) f\tag {2}\\ \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 ) +\left ( \frac {\partial }{\partial y}\frac {d^{2}f}{dx^{2}}\right ) y^{\prime }+\frac {\partial }{\partial y^{\prime }}\left ( \frac {d^{2}f}{dx^{2}}\right ) f\tag {3}\\ & \vdots \nonumber \end {align}

And so on. Hence if we name \(F_{0}=f\left ( x,y,y^{\prime }\right ) \) 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}{\partial x}+\frac {\partial f}{\partial y}y^{\prime }+\frac {\partial f}{\partial y^{\prime }}y^{\prime \prime }\nonumber \\ & =\frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}y^{\prime }+\frac {\partial f}{\partial y^{\prime }}f\tag {5}\\ & =\frac {\partial F_{0}}{\partial x}+\frac {\partial F_{0}}{\partial y}y^{\prime }+\frac {\partial F_{0}}{\partial y^{\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 }{\partial x}F_{1}+\left ( \frac {\partial F_{1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{1}}{\partial y^{\prime }}\right ) y^{\prime \prime }\nonumber \\ & =\frac {\partial }{\partial x}F_{1}+\left ( \frac {\partial F_{1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{1}}{\partial y^{\prime }}\right ) 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 }\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 ) 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 }+\sum _{n=0}^{\infty }\frac {x^{n+2}}{\left ( n+2\right ) !}\left . F_{n}\right \vert _{x_{0},y_{0},y_{0}^{\prime }} \tag {7} \end {equation} To ﬁnd \(y\left ( x\right ) \) series solution around \(x=0\). Hence \begin {align*} F_0 &= \frac {\sin \left (t +{\mathrm e}\right ) y \left (t \right )}{\ln \left (t +{\mathrm e}\right )}\\ F_1 &= \frac {d F_0}{dt} \\ &= \frac {\partial F_{0}}{\partial t}+ \frac {\partial F_{0}}{\partial y} \frac {d}{d t}y \left (t \right )+ \frac {\partial F_{0}}{\partial \frac {d}{d t}y \left (t \right )} F_0 \\ &= \frac {\left (\cos \left (t +{\mathrm e}\right ) y \left (t \right )+\sin \left (t +{\mathrm e}\right ) \left (\frac {d}{d t}y \left (t \right )\right )\right ) \left (t +{\mathrm e}\right ) \ln \left (t +{\mathrm e}\right )-\sin \left (t +{\mathrm e}\right ) y \left (t \right )}{\ln \left (t +{\mathrm e}\right )^{2} \left (t +{\mathrm e}\right )}\\ F_2 &= \frac {d F_1}{dt} \\ &= \frac {\partial F_{1}}{\partial t}+ \frac {\partial F_{1}}{\partial y} \frac {d}{d t}y \left (t \right )+ \frac {\partial F_{1}}{\partial \frac {d}{d t}y \left (t \right )} F_1 \\ &= \frac {-\left (t^{2}+2 \,{\mathrm e} t +{\mathrm e}^{2}\right ) \left (\sin \left (t +{\mathrm e}\right ) y \left (t \right )-2 \cos \left (t +{\mathrm e}\right ) \left (\frac {d}{d t}y \left (t \right )\right )\right ) \ln \left (t +{\mathrm e}\right )^{2}+\left (y \left (t \right ) \left (t^{2}+2 \,{\mathrm e} t +{\mathrm e}^{2}\right ) \sin \left (t +{\mathrm e}\right )^{2}+\left (\left (-2 t -2 \,{\mathrm e}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+y \left (t \right )\right ) \sin \left (t +{\mathrm e}\right )-2 y \left (t \right ) \cos \left (t +{\mathrm e}\right ) \left (t +{\mathrm e}\right )\right ) \ln \left (t +{\mathrm e}\right )+2 \sin \left (t +{\mathrm e}\right ) y \left (t \right )}{\ln \left (t +{\mathrm e}\right )^{3} \left (t +{\mathrm e}\right )^{2}}\\ F_3 &= \frac {d F_2}{dt} \\ &= \frac {\partial F_{2}}{\partial t}+ \frac {\partial F_{2}}{\partial y} \frac {d}{d t}y \left (t \right )+ \frac {\partial F_{2}}{\partial \frac {d}{d t}y \left (t \right )} F_2 \\ &= \frac {-\left (t^{3}+3 t^{2} {\mathrm e}+3 t \,{\mathrm e}^{2}+{\mathrm e}^{3}\right ) \left (\cos \left (t +{\mathrm e}\right ) y \left (t \right )+3 \sin \left (t +{\mathrm e}\right ) \left (\frac {d}{d t}y \left (t \right )\right )\right ) \ln \left (t +{\mathrm e}\right )^{3}+\left (\left (\frac {d}{d t}y \left (t \right )\right ) \left (t^{3}+3 t^{2} {\mathrm e}+3 t \,{\mathrm e}^{2}+{\mathrm e}^{3}\right ) \sin \left (t +{\mathrm e}\right )^{2}+\left (4 y \left (t \right ) \left (t^{3}+3 t^{2} {\mathrm e}+3 t \,{\mathrm e}^{2}+{\mathrm e}^{3}\right ) \cos \left (t +{\mathrm e}\right )+\left (3 t +3 \,{\mathrm e}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+3 \left (t^{2}+2 \,{\mathrm e} t +{\mathrm e}^{2}-\frac {2}{3}\right ) y \left (t \right )\right ) \sin \left (t +{\mathrm e}\right )-6 \left (\left (t^{2}+2 \,{\mathrm e} t +{\mathrm e}^{2}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-\frac {y \left (t \right ) \left (t +{\mathrm e}\right )}{2}\right ) \cos \left (t +{\mathrm e}\right )\right ) \ln \left (t +{\mathrm e}\right )^{2}+\left (-4 y \left (t \right ) \left (t^{2}+2 \,{\mathrm e} t +{\mathrm e}^{2}\right ) \sin \left (t +{\mathrm e}\right )^{2}+\left (\left (6 t +6 \,{\mathrm e}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-6 y \left (t \right )\right ) \sin \left (t +{\mathrm e}\right )+6 y \left (t \right ) \cos \left (t +{\mathrm e}\right ) \left (t +{\mathrm e}\right )\right ) \ln \left (t +{\mathrm e}\right )-6 \sin \left (t +{\mathrm e}\right ) y \left (t \right )}{\ln \left (t +{\mathrm e}\right )^{4} \left (t +{\mathrm e}\right )^{3}}\\ F_4 &= \frac {d F_3}{dt} \\ &= \frac {\partial F_{3}}{\partial t}+ \frac {\partial F_{3}}{\partial y} \frac {d}{d t}y \left (t \right )+ \frac {\partial F_{3}}{\partial \frac {d}{d t}y \left (t \right )} F_3 \\ &= \frac {\left (t^{4}+4 t^{3} {\mathrm e}+6 t^{2} {\mathrm e}^{2}+4 t \,{\mathrm e}^{3}+{\mathrm e}^{4}\right ) \left (\sin \left (t +{\mathrm e}\right ) y \left (t \right )-4 \cos \left (t +{\mathrm e}\right ) \left (\frac {d}{d t}y \left (t \right )\right )\right ) \ln \left (t +{\mathrm e}\right )^{4}+\left (\left (6 \left (\frac {d}{d t}y \left (t \right )\right ) \left (t^{4}+4 t^{3} {\mathrm e}+6 t^{2} {\mathrm e}^{2}+4 t \,{\mathrm e}^{3}+{\mathrm e}^{4}\right ) \cos \left (t +{\mathrm e}\right )+\left (\left (36 t^{2}-8\right ) {\mathrm e}+12 t^{3}+36 t \,{\mathrm e}^{2}-8 t +12 \,{\mathrm e}^{3}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-6 y \left (t \right ) \left (t^{2}+2 \,{\mathrm e} t +{\mathrm e}^{2}-1\right )\right ) \sin \left (t +{\mathrm e}\right )+11 y \left (t \right ) \left (t^{4}+4 t^{3} {\mathrm e}+6 t^{2} {\mathrm e}^{2}+4 t \,{\mathrm e}^{3}+{\mathrm e}^{4}\right ) \cos \left (t +{\mathrm e}\right )^{2}+\left (\left (12 t^{2}+24 \,{\mathrm e} t +12 \,{\mathrm e}^{2}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+4 \left (\left (3 t^{2}-2\right ) {\mathrm e}+t^{3}+3 t \,{\mathrm e}^{2}-2 t +{\mathrm e}^{3}\right ) y \left (t \right )\right ) \cos \left (t +{\mathrm e}\right )-7 y \left (t \right ) \left (t^{4}+4 t^{3} {\mathrm e}+6 t^{2} {\mathrm e}^{2}+4 t \,{\mathrm e}^{3}+{\mathrm e}^{4}\right )\right ) \ln \left (t +{\mathrm e}\right )^{3}+\left (\sin \left (t +{\mathrm e}\right )^{3} y \left (t \right ) \left (t^{4}+4 t^{3} {\mathrm e}+6 t^{2} {\mathrm e}^{2}+4 t \,{\mathrm e}^{3}+{\mathrm e}^{4}\right )+\left (\left (-6 t^{3}-18 t^{2} {\mathrm e}-18 t \,{\mathrm e}^{2}-6 \,{\mathrm e}^{3}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+7 y \left (t \right ) \left (t^{2}+2 \,{\mathrm e} t +{\mathrm e}^{2}\right )\right ) \sin \left (t +{\mathrm e}\right )^{2}+\left (-22 y \left (t \right ) \left (t^{3}+3 t^{2} {\mathrm e}+3 t \,{\mathrm e}^{2}+{\mathrm e}^{3}\right ) \cos \left (t +{\mathrm e}\right )+\left (-24 t -24 \,{\mathrm e}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-12 \left (t^{2}+2 \,{\mathrm e} t +{\mathrm e}^{2}-\frac {11}{6}\right ) y \left (t \right )\right ) \sin \left (t +{\mathrm e}\right )+24 \cos \left (t +{\mathrm e}\right ) \left (\left (t^{2}+2 \,{\mathrm e} t +{\mathrm e}^{2}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-y \left (t \right ) \left (t +{\mathrm e}\right )\right )\right ) \ln \left (t +{\mathrm e}\right )^{2}+\left (18 y \left (t \right ) \left (t^{2}+2 \,{\mathrm e} t +{\mathrm e}^{2}\right ) \sin \left (t +{\mathrm e}\right )^{2}+\left (\left (-24 t -24 \,{\mathrm e}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+36 y \left (t \right )\right ) \sin \left (t +{\mathrm e}\right )-24 y \left (t \right ) \cos \left (t +{\mathrm e}\right ) \left (t +{\mathrm e}\right )\right ) \ln \left (t +{\mathrm e}\right )+24 \sin \left (t +{\mathrm e}\right ) y \left (t \right )}{\ln \left (t +{\mathrm e}\right )^{5} \left (t +{\mathrm e}\right )^{4}} \end {align*}

And so on. Evaluating all the above at initial conditions \(t = 0\) and \(y \left (0\right ) = {\mathrm e}^{-1}\) and \(y^{\prime }\left (0\right ) = 0\) gives \begin {align*} F_0 &= {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )\\ F_1 &= -\sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}+\cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}\\ F_2 &= \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}-2 \,{\mathrm e}^{-2} \cos \left ({\mathrm e}\right )+3 \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}-{\mathrm e}^{-1} \sin \left ({\mathrm e}\right )\\ F_3 &= -4 \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-2}+2 \sin \left (2 \,{\mathrm e}\right ) {\mathrm e}^{-1}+9 \cos \left ({\mathrm e}\right ) {\mathrm e}^{-3}-\cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}+3 \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}-14 \sin \left ({\mathrm e}\right ) {\mathrm e}^{-4}\\ F_4 &= 88 \sin \left ({\mathrm e}\right ) {\mathrm e}^{-5}-56 \cos \left ({\mathrm e}\right ) {\mathrm e}^{-4}+25 \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-3}-18 \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}-11 \sin \left (2 \,{\mathrm e}\right ) {\mathrm e}^{-2}+4 \,{\mathrm e}^{-2} \cos \left ({\mathrm e}\right )-\frac {{\mathrm e}^{-1} \sin \left (3 \,{\mathrm e}\right )}{4}+\frac {7 \,{\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{4}+11 \,{\mathrm e}^{-1} \cos \left ({\mathrm e}\right )^{2}-7 \,{\mathrm e}^{-1} \end {align*}

Substituting all the above in (7) and simplifying gives the solution as \[ y \left (t \right ) = {\mathrm e}^{-1}-\frac {7 t^{6} {\mathrm e}^{-1}}{720}+O\left (t^{6}\right )+\frac {t^{5} \cos \left ({\mathrm e}\right ) \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{30}-\frac {t^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )^{2}}{720}-\frac {11 t^{6} {\mathrm e}^{-2} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{360}+\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) t^{2}}{2}-\frac {t^{3} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{6}+\frac {t^{3} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{6}+\frac {t^{4} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}}{24}-\frac {t^{4} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{12}+\frac {t^{4} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{8}-\frac {t^{4} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{24}-\frac {t^{5} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-2}}{30}+\frac {3 t^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}-\frac {t^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{120}+\frac {t^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{40}-\frac {7 t^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{60}+\frac {11 t^{6} {\mathrm e}^{-1} \cos \left ({\mathrm e}\right )^{2}}{720}+\frac {5 t^{6} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-3}}{144}+\frac {t^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{360}+\frac {t^{6} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{180}-\frac {t^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}+\frac {11 t^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-5}}{90}-\frac {7 t^{6} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{90} \] \[ y \left (t \right ) = {\mathrm e}^{-1}-\frac {7 t^{6} {\mathrm e}^{-1}}{720}+O\left (t^{6}\right )+\frac {t^{5} \cos \left ({\mathrm e}\right ) \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{30}-\frac {t^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )^{2}}{720}-\frac {11 t^{6} {\mathrm e}^{-2} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{360}+\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) t^{2}}{2}-\frac {t^{3} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{6}+\frac {t^{3} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{6}+\frac {t^{4} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}}{24}-\frac {t^{4} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{12}+\frac {t^{4} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{8}-\frac {t^{4} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{24}-\frac {t^{5} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-2}}{30}+\frac {3 t^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}-\frac {t^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{120}+\frac {t^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{40}-\frac {7 t^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{60}+\frac {11 t^{6} {\mathrm e}^{-1} \cos \left ({\mathrm e}\right )^{2}}{720}+\frac {5 t^{6} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-3}}{144}+\frac {t^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{360}+\frac {t^{6} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{180}-\frac {t^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}+\frac {11 t^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-5}}{90}-\frac {7 t^{6} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{90} \] Since the expansion point \(t = 0\) is an ordinary, we can also solve this using standard power series Let the solution be represented as power series of the form \[ y \left (t \right ) = \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n} \] Then \begin {align*} \frac {d}{d t}y \left (t \right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\\ \frac {d^{2}}{d t^{2}}y \left (t \right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2} \end {align*}

Substituting the above back into the ode gives \begin {align*} \moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2} = \frac {\sin \left (t +{\mathrm e}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right )}{\ln \left (t +{\mathrm e}\right )}\tag {1} \end {align*}

Expanding \(\ln \left (t +{\mathrm e}\right )\) as Taylor series around \(t=0\) and keeping only the ﬁrst \(6\) terms gives \begin {align*} \ln \left (t +{\mathrm e}\right ) &= 1+{\mathrm e}^{-1} t -\frac {{\mathrm e}^{-2} t^{2}}{2}+\frac {{\mathrm e}^{-3} t^{3}}{3}-\frac {{\mathrm e}^{-4} t^{4}}{4}+\frac {{\mathrm e}^{-5} t^{5}}{5}-\frac {{\mathrm e}^{-6} t^{6}}{6} + \dots \\ &= 1+{\mathrm e}^{-1} t -\frac {{\mathrm e}^{-2} t^{2}}{2}+\frac {{\mathrm e}^{-3} t^{3}}{3}-\frac {{\mathrm e}^{-4} t^{4}}{4}+\frac {{\mathrm e}^{-5} t^{5}}{5}-\frac {{\mathrm e}^{-6} t^{6}}{6} \end {align*}

Expanding \(-\sin \left (t +{\mathrm e}\right )\) as Taylor series around \(t=0\) and keeping only the ﬁrst \(6\) terms gives \begin {align*} -\sin \left (t +{\mathrm e}\right ) &= -\sin \left ({\mathrm e}\right )-\cos \left ({\mathrm e}\right ) t +\frac {\sin \left ({\mathrm e}\right ) t^{2}}{2}+\frac {\cos \left ({\mathrm e}\right ) t^{3}}{6}-\frac {\sin \left ({\mathrm e}\right ) t^{4}}{24}-\frac {\cos \left ({\mathrm e}\right ) t^{5}}{120}+\frac {\sin \left ({\mathrm e}\right ) t^{6}}{720} + \dots \\ &= -\sin \left ({\mathrm e}\right )-\cos \left ({\mathrm e}\right ) t +\frac {\sin \left ({\mathrm e}\right ) t^{2}}{2}+\frac {\cos \left ({\mathrm e}\right ) t^{3}}{6}-\frac {\sin \left ({\mathrm e}\right ) t^{4}}{24}-\frac {\cos \left ({\mathrm e}\right ) t^{5}}{120}+\frac {\sin \left ({\mathrm e}\right ) t^{6}}{720} \end {align*}

Hence the ODE in Eq (1) becomes \[ \left (1+{\mathrm e}^{-1} t -\frac {{\mathrm e}^{-2} t^{2}}{2}+\frac {{\mathrm e}^{-3} t^{3}}{3}-\frac {{\mathrm e}^{-4} t^{4}}{4}+\frac {{\mathrm e}^{-5} t^{5}}{5}-\frac {{\mathrm e}^{-6} t^{6}}{6}\right ) \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\left (-\sin \left ({\mathrm e}\right )-\cos \left ({\mathrm e}\right ) t +\frac {\sin \left ({\mathrm e}\right ) t^{2}}{2}+\frac {\cos \left ({\mathrm e}\right ) t^{3}}{6}-\frac {\sin \left ({\mathrm e}\right ) t^{4}}{24}-\frac {\cos \left ({\mathrm e}\right ) t^{5}}{120}+\frac {\sin \left ({\mathrm e}\right ) t^{6}}{720}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right ) = 0 \] Expanding the ﬁrst term in (1) gives \[ \left (1\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )\right )+{\mathrm e}^{-1} t \eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )-\frac {{\mathrm e}^{-2} t^{2}}{2}\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {{\mathrm e}^{-3} t^{3}}{3}\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )-\frac {{\mathrm e}^{-4} t^{4}}{4}\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {{\mathrm e}^{-5} t^{5}}{5}\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )-\frac {{\mathrm e}^{-6} t^{6}}{6}\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\left (-\sin \left ({\mathrm e}\right )-\cos \left ({\mathrm e}\right ) t +\frac {\sin \left ({\mathrm e}\right ) t^{2}}{2}+\frac {\cos \left ({\mathrm e}\right ) t^{3}}{6}-\frac {\sin \left ({\mathrm e}\right ) t^{4}}{24}-\frac {\cos \left ({\mathrm e}\right ) t^{5}}{120}+\frac {\sin \left ({\mathrm e}\right ) t^{6}}{720}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right ) = 0 \] Expanding the second term in (1) gives \[ \left (1\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )\right )+{\mathrm e}^{-1} t \eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )-\frac {{\mathrm e}^{-2} t^{2}}{2}\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {{\mathrm e}^{-3} t^{3}}{3}\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )-\frac {{\mathrm e}^{-4} t^{4}}{4}\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {{\mathrm e}^{-5} t^{5}}{5}\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )-\frac {{\mathrm e}^{-6} t^{6}}{6}\eslowast \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )-\sin \left ({\mathrm e}\right )\eslowast \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right )-\cos \left ({\mathrm e}\right ) t \eslowast \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right )+\frac {\sin \left ({\mathrm e}\right ) t^{2}}{2}\eslowast \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right )+\frac {\cos \left ({\mathrm e}\right ) t^{3}}{6}\eslowast \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right )-\frac {\sin \left ({\mathrm e}\right ) t^{4}}{24}\eslowast \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right )-\frac {\cos \left ({\mathrm e}\right ) t^{5}}{120}\eslowast \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right )+\frac {\sin \left ({\mathrm e}\right ) t^{6}}{720}\eslowast \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right ) = 0 \] Which simpliﬁes to \begin{equation} \tag{2} \moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n \,t^{n +4} a_{n} \left (n -1\right ) {\mathrm e}^{-6}}{6}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +3} a_{n} \left (n -1\right ) {\mathrm e}^{-5}}{5}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n \,t^{n +2} a_{n} \left (n -1\right ) {\mathrm e}^{-4}}{4}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{1+n} a_{n} \left (n -1\right ) {\mathrm e}^{-3}}{3}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n a_{n} t^{n} \left (n -1\right ) {\mathrm e}^{-2}}{2}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}n \,t^{n -1} a_{n} \left (n -1\right ) {\mathrm e}^{-1}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-a_{n} t^{n} \sin \left ({\mathrm e}\right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-t^{1+n} a_{n} \cos \left ({\mathrm e}\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +2} a_{n} \sin \left ({\mathrm e}\right )}{2}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +3} a_{n} \cos \left ({\mathrm e}\right )}{6}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +4} a_{n} \sin \left ({\mathrm e}\right )}{24}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +5} a_{n} \cos \left ({\mathrm e}\right )}{120}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +6} a_{n} \sin \left ({\mathrm e}\right )}{720}\right ) = 0 \end{equation} The next step is to make all powers of \(t\) be \(n\) in each summation term. Going over each summation term above with power of \(t\) in it which is not already \(t^{n}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n \,t^{n +4} a_{n} \left (n -1\right ) {\mathrm e}^{-6}}{6}\right ) &= \moverset {\infty }{\munderset {n =6}{\sum }}\left (-\frac {\left (n -4\right ) a_{n -4} \left (n -5\right ) {\mathrm e}^{-6} t^{n}}{6}\right ) \\ \moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +3} a_{n} \left (n -1\right ) {\mathrm e}^{-5}}{5} &= \moverset {\infty }{\munderset {n =5}{\sum }}\frac {\left (n -3\right ) a_{n -3} \left (n -4\right ) {\mathrm e}^{-5} t^{n}}{5} \\ \moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n \,t^{n +2} a_{n} \left (n -1\right ) {\mathrm e}^{-4}}{4}\right ) &= \moverset {\infty }{\munderset {n =4}{\sum }}\left (-\frac {\left (n -2\right ) a_{n -2} \left (n -3\right ) {\mathrm e}^{-4} t^{n}}{4}\right ) \\ \moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{1+n} a_{n} \left (n -1\right ) {\mathrm e}^{-3}}{3} &= \moverset {\infty }{\munderset {n =3}{\sum }}\frac {\left (n -1\right ) a_{n -1} \left (n -2\right ) {\mathrm e}^{-3} t^{n}}{3} \\ \moverset {\infty }{\munderset {n =2}{\sum }}n \,t^{n -1} a_{n} \left (n -1\right ) {\mathrm e}^{-1} &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (1+n \right ) a_{1+n} n \,{\mathrm e}^{-1} t^{n} \\ \moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2} &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +2\right ) a_{n +2} \left (1+n \right ) t^{n} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-t^{1+n} a_{n} \cos \left ({\mathrm e}\right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \cos \left ({\mathrm e}\right ) t^{n}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +2} a_{n} \sin \left ({\mathrm e}\right )}{2} &= \moverset {\infty }{\munderset {n =2}{\sum }}\frac {a_{n -2} \sin \left ({\mathrm e}\right ) t^{n}}{2} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +3} a_{n} \cos \left ({\mathrm e}\right )}{6} &= \moverset {\infty }{\munderset {n =3}{\sum }}\frac {a_{n -3} \cos \left ({\mathrm e}\right ) t^{n}}{6} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +4} a_{n} \sin \left ({\mathrm e}\right )}{24}\right ) &= \moverset {\infty }{\munderset {n =4}{\sum }}\left (-\frac {a_{n -4} \sin \left ({\mathrm e}\right ) t^{n}}{24}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +5} a_{n} \cos \left ({\mathrm e}\right )}{120}\right ) &= \moverset {\infty }{\munderset {n =5}{\sum }}\left (-\frac {a_{n -5} \cos \left ({\mathrm e}\right ) t^{n}}{120}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +6} a_{n} \sin \left ({\mathrm e}\right )}{720} &= \moverset {\infty }{\munderset {n =6}{\sum }}\frac {a_{n -6} \sin \left ({\mathrm e}\right ) t^{n}}{720} \\ \end{align*} Substituting all the above in Eq (2) gives the following equation where now all powers of \(t\) are the same and equal to \(n\). \begin{equation} \tag{3} \moverset {\infty }{\munderset {n =6}{\sum }}\left (-\frac {\left (n -4\right ) a_{n -4} \left (n -5\right ) {\mathrm e}^{-6} t^{n}}{6}\right )+\left (\moverset {\infty }{\munderset {n =5}{\sum }}\frac {\left (n -3\right ) a_{n -3} \left (n -4\right ) {\mathrm e}^{-5} t^{n}}{5}\right )+\moverset {\infty }{\munderset {n =4}{\sum }}\left (-\frac {\left (n -2\right ) a_{n -2} \left (n -3\right ) {\mathrm e}^{-4} t^{n}}{4}\right )+\left (\moverset {\infty }{\munderset {n =3}{\sum }}\frac {\left (n -1\right ) a_{n -1} \left (n -2\right ) {\mathrm e}^{-3} t^{n}}{3}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n a_{n} t^{n} \left (n -1\right ) {\mathrm e}^{-2}}{2}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}\left (1+n \right ) a_{1+n} n \,{\mathrm e}^{-1} t^{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +2\right ) a_{n +2} \left (1+n \right ) t^{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-a_{n} t^{n} \sin \left ({\mathrm e}\right )\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \cos \left ({\mathrm e}\right ) t^{n}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {a_{n -2} \sin \left ({\mathrm e}\right ) t^{n}}{2}\right )+\left (\moverset {\infty }{\munderset {n =3}{\sum }}\frac {a_{n -3} \cos \left ({\mathrm e}\right ) t^{n}}{6}\right )+\moverset {\infty }{\munderset {n =4}{\sum }}\left (-\frac {a_{n -4} \sin \left ({\mathrm e}\right ) t^{n}}{24}\right )+\moverset {\infty }{\munderset {n =5}{\sum }}\left (-\frac {a_{n -5} \cos \left ({\mathrm e}\right ) t^{n}}{120}\right )+\left (\moverset {\infty }{\munderset {n =6}{\sum }}\frac {a_{n -6} \sin \left ({\mathrm e}\right ) t^{n}}{720}\right ) = 0 \end{equation} \(n=0\) gives \[ 2 a_{2}-a_{0} \sin \left ({\mathrm e}\right )=0 \] \[ a_{2} = \frac {a_{0} \sin \left ({\mathrm e}\right )}{2} \] \(n=1\) gives \[ 2 a_{2} {\mathrm e}^{-1}+6 a_{3}-a_{1} \sin \left ({\mathrm e}\right )-a_{0} \cos \left ({\mathrm e}\right )=0 \] Which after substituting earlier equations, simpliﬁes to \[ a_{3} = -\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{6}+\frac {a_{0} \cos \left ({\mathrm e}\right )}{6}+\frac {a_{1} \sin \left ({\mathrm e}\right )}{6} \] \(n=2\) gives \[ -a_{2} {\mathrm e}^{-2}+6 a_{3} {\mathrm e}^{-1}+12 a_{4}-a_{2} \sin \left ({\mathrm e}\right )-a_{1} \cos \left ({\mathrm e}\right )+\frac {a_{0} \sin \left ({\mathrm e}\right )}{2}=0 \] Which after substituting earlier equations, simpliﬁes to \[ a_{4} = \frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{8}-\frac {{\mathrm e}^{-1} a_{0} \cos \left ({\mathrm e}\right )}{12}+\frac {a_{0} \sin \left ({\mathrm e}\right )^{2}}{24}-\frac {a_{0} \sin \left ({\mathrm e}\right )}{24}-\frac {{\mathrm e}^{-1} a_{1} \sin \left ({\mathrm e}\right )}{12}+\frac {a_{1} \cos \left ({\mathrm e}\right )}{12} \] \(n=3\) gives \[ \frac {2 a_{2} {\mathrm e}^{-3}}{3}-3 a_{3} {\mathrm e}^{-2}+12 a_{4} {\mathrm e}^{-1}+20 a_{5}-a_{3} \sin \left ({\mathrm e}\right )-a_{2} \cos \left ({\mathrm e}\right )+\frac {a_{1} \sin \left ({\mathrm e}\right )}{2}+\frac {a_{0} \cos \left ({\mathrm e}\right )}{6}=0 \] Which after substituting earlier equations, simpliﬁes to \[ a_{5} = -\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{15}-\frac {{\mathrm e}^{-2} a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{20}+\frac {3 \,{\mathrm e}^{-2} a_{0} \cos \left ({\mathrm e}\right )}{40}-\frac {{\mathrm e}^{-1} a_{0} \sin \left ({\mathrm e}\right )^{2}}{30}+\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{40}+\frac {a_{0} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{30}-\frac {a_{0} \cos \left ({\mathrm e}\right )}{120}+\frac {3 \,{\mathrm e}^{-2} a_{1} \sin \left ({\mathrm e}\right )}{40}-\frac {{\mathrm e}^{-1} a_{1} \cos \left ({\mathrm e}\right )}{20}+\frac {a_{1} \sin \left ({\mathrm e}\right )^{2}}{120}-\frac {a_{1} \sin \left ({\mathrm e}\right )}{40} \] \(n=4\) gives \[ -\frac {a_{2} {\mathrm e}^{-4}}{2}+2 a_{3} {\mathrm e}^{-3}-6 a_{4} {\mathrm e}^{-2}+20 a_{5} {\mathrm e}^{-1}+30 a_{6}-a_{4} \sin \left ({\mathrm e}\right )-a_{3} \cos \left ({\mathrm e}\right )+\frac {a_{2} \sin \left ({\mathrm e}\right )}{2}+\frac {a_{1} \cos \left ({\mathrm e}\right )}{6}-\frac {a_{0} \sin \left ({\mathrm e}\right )}{24}=0 \] Which after substituting earlier equations, simpliﬁes to \[ a_{6} = \frac {a_{0} \sin \left ({\mathrm e}\right )^{3}}{720}+\frac {a_{0} \cos \left ({\mathrm e}\right )^{2}}{180}+\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{20}+\frac {{\mathrm e}^{-3} a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{45}-\frac {{\mathrm e}^{-2} {\mathrm e}^{-1} a_{0} \cos \left ({\mathrm e}\right )}{30}-\frac {{\mathrm e}^{-2} {\mathrm e}^{-1} a_{1} \sin \left ({\mathrm e}\right )}{30}-\frac {11 \,{\mathrm e}^{-1} a_{0} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{360}-\frac {7 a_{0} \sin \left ({\mathrm e}\right )^{2}}{720}-\frac {a_{1} \cos \left ({\mathrm e}\right )}{180}+\frac {a_{0} \sin \left ({\mathrm e}\right )}{720}-\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{40}+\frac {{\mathrm e}^{-1} a_{0} \cos \left ({\mathrm e}\right )}{180}+\frac {{\mathrm e}^{-1} a_{1} \sin \left ({\mathrm e}\right )}{60}-\frac {2 \,{\mathrm e}^{-3} a_{0} \cos \left ({\mathrm e}\right )}{45}-\frac {2 \,{\mathrm e}^{-3} a_{1} \sin \left ({\mathrm e}\right )}{45}+\frac {a_{0} \sin \left ({\mathrm e}\right ) \left ({\mathrm e}^{-2}\right )^{2}}{20}+\frac {5 \,{\mathrm e}^{-2} a_{0} \sin \left ({\mathrm e}\right )^{2}}{144}+\frac {{\mathrm e}^{-2} a_{1} \cos \left ({\mathrm e}\right )}{20}-\frac {{\mathrm e}^{-1} a_{1} \sin \left ({\mathrm e}\right )^{2}}{120}+\frac {\sin \left ({\mathrm e}\right ) a_{1} \cos \left ({\mathrm e}\right )}{120} \] \(n=5\) gives \[ \frac {2 a_{2} {\mathrm e}^{-5}}{5}-\frac {3 a_{3} {\mathrm e}^{-4}}{2}+4 a_{4} {\mathrm e}^{-3}-10 a_{5} {\mathrm e}^{-2}+30 a_{6} {\mathrm e}^{-1}+42 a_{7}-a_{5} \sin \left ({\mathrm e}\right )-a_{4} \cos \left ({\mathrm e}\right )+\frac {a_{3} \sin \left ({\mathrm e}\right )}{2}+\frac {a_{2} \cos \left ({\mathrm e}\right )}{6}-\frac {a_{1} \sin \left ({\mathrm e}\right )}{24}-\frac {a_{0} \cos \left ({\mathrm e}\right )}{120}=0 \] Which after substituting earlier equations, simpliﬁes to \[ a_{7} = \frac {a_{1} \sin \left ({\mathrm e}\right )^{3}}{5040}+\frac {a_{1} \cos \left ({\mathrm e}\right )^{2}}{504}-\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-5}}{35}-\frac {5 \,{\mathrm e}^{-4} a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{168}-\frac {5 \,{\mathrm e}^{-3} a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{63}+\frac {{\mathrm e}^{-3} {\mathrm e}^{-1} a_{0} \cos \left ({\mathrm e}\right )}{63}+\frac {{\mathrm e}^{-3} {\mathrm e}^{-1} a_{1} \sin \left ({\mathrm e}\right )}{63}-\frac {9 \,{\mathrm e}^{-2} {\mathrm e}^{-1} a_{0} \sin \left ({\mathrm e}\right )^{2}}{560}+\frac {29 \,{\mathrm e}^{-2} a_{0} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{840}-\frac {{\mathrm e}^{-2} {\mathrm e}^{-1} a_{1} \cos \left ({\mathrm e}\right )}{42}-\frac {23 \,{\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) a_{1} \cos \left ({\mathrm e}\right )}{2520}+\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{63}-\frac {13 a_{0} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{2520}-\frac {{\mathrm e}^{-2} a_{0} \cos \left ({\mathrm e}\right )}{168}-\frac {{\mathrm e}^{-2} a_{1} \sin \left ({\mathrm e}\right )}{56}+\frac {{\mathrm e}^{-1} a_{0} \sin \left ({\mathrm e}\right )^{2}}{105}+\frac {{\mathrm e}^{-1} a_{1} \cos \left ({\mathrm e}\right )}{252}+\frac {a_{0} \cos \left ({\mathrm e}\right )}{5040}+\frac {a_{1} \sin \left ({\mathrm e}\right )}{1008}+\frac {{\mathrm e}^{-2} a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{84}-\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{1008}-\frac {13 a_{1} \sin \left ({\mathrm e}\right )^{2}}{5040}+\frac {{\mathrm e}^{-4} a_{0} \cos \left ({\mathrm e}\right )}{28}+\frac {{\mathrm e}^{-4} a_{1} \sin \left ({\mathrm e}\right )}{28}-\frac {59 \,{\mathrm e}^{-3} a_{0} \sin \left ({\mathrm e}\right )^{2}}{2520}-\frac {2 \,{\mathrm e}^{-3} a_{1} \cos \left ({\mathrm e}\right )}{63}+\frac {\left ({\mathrm e}^{-2}\right )^{2} a_{0} \cos \left ({\mathrm e}\right )}{28}+\frac {\left ({\mathrm e}^{-2}\right )^{2} a_{1} \sin \left ({\mathrm e}\right )}{28}+\frac {7 \,{\mathrm e}^{-2} a_{1} \sin \left ({\mathrm e}\right )^{2}}{720}-\frac {{\mathrm e}^{-1} a_{0} \sin \left ({\mathrm e}\right )^{3}}{560}-\frac {{\mathrm e}^{-1} a_{0} \cos \left ({\mathrm e}\right )^{2}}{168}+\frac {a_{0} \sin \left ({\mathrm e}\right )^{2} \cos \left ({\mathrm e}\right )}{560} \] For \(6\le n\), the recurrence equation is \begin{equation} \tag{4} -\frac {\left (n -4\right ) a_{n -4} \left (n -5\right ) {\mathrm e}^{-6}}{6}+\frac {\left (n -3\right ) a_{n -3} \left (n -4\right ) {\mathrm e}^{-5}}{5}-\frac {\left (n -2\right ) a_{n -2} \left (n -3\right ) {\mathrm e}^{-4}}{4}+\frac {\left (n -1\right ) a_{n -1} \left (n -2\right ) {\mathrm e}^{-3}}{3}-\frac {n a_{n} \left (n -1\right ) {\mathrm e}^{-2}}{2}+\left (1+n \right ) a_{1+n} n \,{\mathrm e}^{-1}+\left (n +2\right ) a_{n +2} \left (1+n \right )-a_{n} \sin \left ({\mathrm e}\right )-a_{n -1} \cos \left ({\mathrm e}\right )+\frac {a_{n -2} \sin \left ({\mathrm e}\right )}{2}+\frac {a_{n -3} \cos \left ({\mathrm e}\right )}{6}-\frac {a_{n -4} \sin \left ({\mathrm e}\right )}{24}-\frac {a_{n -5} \cos \left ({\mathrm e}\right )}{120}+\frac {a_{n -6} \sin \left ({\mathrm e}\right )}{720} = 0 \end{equation} Solving for \(a_{n +2}\), gives \begin{align*} \tag{5} a_{n +2}&= \frac {-720 a_{1+n} n^{2} {\mathrm e}^{-1}-144 \,{\mathrm e}^{-5} n^{2} a_{n -3}-240 \,{\mathrm e}^{-3} n^{2} a_{n -1}+120 \,{\mathrm e}^{-6} n^{2} a_{n -4}+360 n^{2} a_{n} {\mathrm e}^{-2}+180 \,{\mathrm e}^{-4} n^{2} a_{n -2}-720 a_{1+n} n \,{\mathrm e}^{-1}+1008 \,{\mathrm e}^{-5} n a_{n -3}+720 \,{\mathrm e}^{-3} n a_{n -1}-1080 \,{\mathrm e}^{-6} n a_{n -4}-360 n a_{n} {\mathrm e}^{-2}-900 \,{\mathrm e}^{-4} n a_{n -2}+6 a_{n -5} \cos \left ({\mathrm e}\right )-120 a_{n -3} \cos \left ({\mathrm e}\right )+720 a_{n -1} \cos \left ({\mathrm e}\right )+720 a_{n} \sin \left ({\mathrm e}\right )-a_{n -6} \sin \left ({\mathrm e}\right )+30 a_{n -4} \sin \left ({\mathrm e}\right )-360 a_{n -2} \sin \left ({\mathrm e}\right )-1728 a_{n -3} {\mathrm e}^{-5}-480 a_{n -1} {\mathrm e}^{-3}+2400 a_{n -4} {\mathrm e}^{-6}+1080 a_{n -2} {\mathrm e}^{-4}}{720 \left (n +2\right ) \left (1+n \right )} \\ &= \frac {\left (360 \,{\mathrm e}^{-2} n^{2}-360 \,{\mathrm e}^{-2} n +720 \sin \left ({\mathrm e}\right )\right ) a_{n}}{720 \left (n +2\right ) \left (1+n \right )}+\frac {\left (-720 \,{\mathrm e}^{-1} n^{2}-720 \,{\mathrm e}^{-1} n \right ) a_{1+n}}{720 \left (n +2\right ) \left (1+n \right )}-\frac {\sin \left ({\mathrm e}\right ) a_{n -6}}{720 \left (n +2\right ) \left (1+n \right )}+\frac {\cos \left ({\mathrm e}\right ) a_{n -5}}{120 \left (n +2\right ) \left (1+n \right )}+\frac {\left (120 \,{\mathrm e}^{-6} n^{2}-1080 \,{\mathrm e}^{-6} n +30 \sin \left ({\mathrm e}\right )+2400 \,{\mathrm e}^{-6}\right ) a_{n -4}}{720 \left (n +2\right ) \left (1+n \right )}+\frac {\left (-144 \,{\mathrm e}^{-5} n^{2}+1008 \,{\mathrm e}^{-5} n -120 \cos \left ({\mathrm e}\right )-1728 \,{\mathrm e}^{-5}\right ) a_{n -3}}{720 \left (n +2\right ) \left (1+n \right )}+\frac {\left (180 \,{\mathrm e}^{-4} n^{2}-900 \,{\mathrm e}^{-4} n -360 \sin \left ({\mathrm e}\right )+1080 \,{\mathrm e}^{-4}\right ) a_{n -2}}{720 \left (n +2\right ) \left (1+n \right )}+\frac {\left (-240 \,{\mathrm e}^{-3} n^{2}+720 \,{\mathrm e}^{-3} n +720 \cos \left ({\mathrm e}\right )-480 \,{\mathrm e}^{-3}\right ) a_{n -1}}{720 \left (n +2\right ) \left (1+n \right )} \\ \end{align*} And so on. Therefore the solution is \begin {align*} y \left (t \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\\ &= a_{3} t^{3}+a_{2} t^{2}+a_{1} t +a_{0} + \dots \end {align*}

Substituting the values for \(a_{n}\) found above, the solution becomes \[ y \left (t \right ) = a_{0}+a_{1} t +\frac {a_{0} \sin \left ({\mathrm e}\right ) t^{2}}{2}+\left (-\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{6}+\frac {a_{0} \cos \left ({\mathrm e}\right )}{6}+\frac {a_{1} \sin \left ({\mathrm e}\right )}{6}\right ) t^{3}+\left (\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{8}-\frac {{\mathrm e}^{-1} a_{0} \cos \left ({\mathrm e}\right )}{12}+\frac {a_{0} \sin \left ({\mathrm e}\right )^{2}}{24}-\frac {a_{0} \sin \left ({\mathrm e}\right )}{24}-\frac {{\mathrm e}^{-1} a_{1} \sin \left ({\mathrm e}\right )}{12}+\frac {a_{1} \cos \left ({\mathrm e}\right )}{12}\right ) t^{4}+\left (-\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{15}-\frac {{\mathrm e}^{-2} a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{20}+\frac {3 \,{\mathrm e}^{-2} a_{0} \cos \left ({\mathrm e}\right )}{40}-\frac {{\mathrm e}^{-1} a_{0} \sin \left ({\mathrm e}\right )^{2}}{30}+\frac {a_{0} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{40}+\frac {a_{0} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{30}-\frac {a_{0} \cos \left ({\mathrm e}\right )}{120}+\frac {3 \,{\mathrm e}^{-2} a_{1} \sin \left ({\mathrm e}\right )}{40}-\frac {{\mathrm e}^{-1} a_{1} \cos \left ({\mathrm e}\right )}{20}+\frac {a_{1} \sin \left ({\mathrm e}\right )^{2}}{120}-\frac {a_{1} \sin \left ({\mathrm e}\right )}{40}\right ) t^{5}+\dots \] Collecting terms, the solution becomes \begin{equation} \tag{3} y \left (t \right ) = \left (1+\frac {\sin \left ({\mathrm e}\right ) t^{2}}{2}+\left (-\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{6}+\frac {\cos \left ({\mathrm e}\right )}{6}\right ) t^{3}+\left (\frac {\sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{8}-\frac {\cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{12}+\frac {\sin \left ({\mathrm e}\right )^{2}}{24}-\frac {\sin \left ({\mathrm e}\right )}{24}\right ) t^{4}+\left (-\frac {\sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{15}-\frac {{\mathrm e}^{-2} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{20}+\frac {3 \,{\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{40}-\frac {\sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}}{30}+\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{40}+\frac {\sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{30}-\frac {\cos \left ({\mathrm e}\right )}{120}\right ) t^{5}\right ) a_{0}+\left (t +\frac {\sin \left ({\mathrm e}\right ) t^{3}}{6}+\left (-\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{12}+\frac {\cos \left ({\mathrm e}\right )}{12}\right ) t^{4}+\left (\frac {3 \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{40}-\frac {\cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{20}+\frac {\sin \left ({\mathrm e}\right )^{2}}{120}-\frac {\sin \left ({\mathrm e}\right )}{40}\right ) t^{5}\right ) a_{1}+O\left (t^{6}\right ) \end{equation} At \(t = 0\) the solution above becomes \[ y \left (t \right ) = \left (1+\frac {\sin \left ({\mathrm e}\right ) t^{2}}{2}+\left (-\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{6}+\frac {\cos \left ({\mathrm e}\right )}{6}\right ) t^{3}+\left (\frac {\sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{8}-\frac {\cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{12}+\frac {\sin \left ({\mathrm e}\right )^{2}}{24}-\frac {\sin \left ({\mathrm e}\right )}{24}\right ) t^{4}+\left (-\frac {\sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{15}-\frac {{\mathrm e}^{-2} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{20}+\frac {3 \,{\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{40}-\frac {\sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}}{30}+\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{40}+\frac {\sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{30}-\frac {\cos \left ({\mathrm e}\right )}{120}\right ) t^{5}\right ) c_{1} +\left (t +\frac {\sin \left ({\mathrm e}\right ) t^{3}}{6}+\left (-\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{12}+\frac {\cos \left ({\mathrm e}\right )}{12}\right ) t^{4}+\left (\frac {3 \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{40}-\frac {\cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{20}+\frac {\sin \left ({\mathrm e}\right )^{2}}{120}-\frac {\sin \left ({\mathrm e}\right )}{40}\right ) t^{5}\right ) c_{2} +O\left (t^{6}\right ) \] \[ y \left (t \right ) = {\mathrm e}^{-1}+\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) t^{2}}{2}-\frac {\left ({\mathrm e}^{-1}\right )^{2} t^{3} \sin \left ({\mathrm e}\right )}{6}+\frac {t^{3} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{6}+\frac {{\mathrm e}^{-1} t^{4} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{8}-\frac {\left ({\mathrm e}^{-1}\right )^{2} t^{4} \cos \left ({\mathrm e}\right )}{12}+\frac {t^{4} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}}{24}-\frac {t^{4} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{24}-\frac {{\mathrm e}^{-1} t^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{15}-\frac {\left ({\mathrm e}^{-1}\right )^{2} t^{5} {\mathrm e}^{-2} \sin \left ({\mathrm e}\right )}{20}+\frac {3 \,{\mathrm e}^{-1} t^{5} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{40}-\frac {\left ({\mathrm e}^{-1}\right )^{2} t^{5} \sin \left ({\mathrm e}\right )^{2}}{30}+\frac {\left ({\mathrm e}^{-1}\right )^{2} t^{5} \sin \left ({\mathrm e}\right )}{40}+\frac {t^{5} \cos \left ({\mathrm e}\right ) \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{30}-\frac {t^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{120}+O\left (t^{6}\right ) \] Replacing \(t\) in the above with the original independent variable \(xs\)using \(t = x -{\mathrm e}\) results in \[ y = {\mathrm e}^{-1}-\frac {\left (x -{\mathrm e}\right )^{3} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{6}+\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) \left (x -{\mathrm e}\right )^{2}}{2}+\frac {\left (x -{\mathrm e}\right )^{3} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{6}+\frac {\left (x -{\mathrm e}\right )^{4} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}}{24}-\frac {\left (x -{\mathrm e}\right )^{4} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{12}+\frac {\left (x -{\mathrm e}\right )^{4} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{8}-\frac {\left (x -{\mathrm e}\right )^{4} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{24}-\frac {\left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-2}}{30}+\frac {3 \left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}-\frac {\left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{120}+\frac {\left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{40}-\frac {7 \left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{60}+\frac {11 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \cos \left ({\mathrm e}\right )^{2}}{720}+\frac {5 \left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-3}}{144}+\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{360}+\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{180}-\frac {\left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}+\frac {11 \left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-5}}{90}-\frac {7 \left (x -{\mathrm e}\right )^{6} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{90}+O\left (\left (x -{\mathrm e}\right )^{6}\right )-\frac {7 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1}}{720}+\frac {\left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{30}-\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )^{2}}{720}-\frac {11 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-2} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{360} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-1}-\frac {\left (x -{\mathrm e}\right )^{3} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{6}+\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) \left (x -{\mathrm e}\right )^{2}}{2}+\frac {\left (x -{\mathrm e}\right )^{3} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{6}+\frac {\left (x -{\mathrm e}\right )^{4} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}}{24}-\frac {\left (x -{\mathrm e}\right )^{4} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{12}+\frac {\left (x -{\mathrm e}\right )^{4} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{8}-\frac {\left (x -{\mathrm e}\right )^{4} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{24}-\frac {\left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-2}}{30}+\frac {3 \left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}-\frac {\left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{120}+\frac {\left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{40}-\frac {7 \left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{60}+\frac {11 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \cos \left ({\mathrm e}\right )^{2}}{720}+\frac {5 \left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-3}}{144}+\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{360}+\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{180}-\frac {\left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}+\frac {11 \left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-5}}{90}-\frac {7 \left (x -{\mathrm e}\right )^{6} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{90}+O\left (\left (x -{\mathrm e}\right )^{6}\right )-\frac {7 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1}}{720}+\frac {\left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{30}-\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )^{2}}{720}-\frac {11 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-2} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{360} \\ \end{align*}

Veriﬁcation of solutions

\[ y = {\mathrm e}^{-1}-\frac {\left (x -{\mathrm e}\right )^{3} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{6}+\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) \left (x -{\mathrm e}\right )^{2}}{2}+\frac {\left (x -{\mathrm e}\right )^{3} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{6}+\frac {\left (x -{\mathrm e}\right )^{4} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}}{24}-\frac {\left (x -{\mathrm e}\right )^{4} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{12}+\frac {\left (x -{\mathrm e}\right )^{4} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{8}-\frac {\left (x -{\mathrm e}\right )^{4} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{24}-\frac {\left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-2}}{30}+\frac {3 \left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}-\frac {\left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{120}+\frac {\left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{40}-\frac {7 \left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{60}+\frac {11 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \cos \left ({\mathrm e}\right )^{2}}{720}+\frac {5 \left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-3}}{144}+\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{360}+\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{180}-\frac {\left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}+\frac {11 \left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-5}}{90}-\frac {7 \left (x -{\mathrm e}\right )^{6} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{90}+O\left (\left (x -{\mathrm e}\right )^{6}\right )-\frac {7 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1}}{720}+\frac {\left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{30}-\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )^{2}}{720}-\frac {11 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-2} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right )}{360} \] Veriﬁed OK.

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      trying 2nd order exact linear 
      trying symmetries linear in x and y(x) 
      trying to convert to a linear ODE with constant coefficients 
      -> trying with_periodic_functions in the coefficients 
         --- Trying Lie symmetry methods, 2nd order --- 
         `, `-> Computing symmetries using: way = 5`[0, u]

\[ y \left (x \right ) = {\mathrm e}^{-1}+\frac {1}{2} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1} \left (x -{\mathrm e}\right )^{2}+\frac {1}{6} \left (\cos \left ({\mathrm e}\right ) {\mathrm e}-\sin \left ({\mathrm e}\right )\right ) {\mathrm e}^{-2} \left (x -{\mathrm e}\right )^{3}+\left (\frac {{\mathrm e}^{-3} {\mathrm e}^{2} \sin \left ({\mathrm e}\right )^{2}}{24}-\frac {\left ({\mathrm e}^{2}-3\right ) {\mathrm e}^{-3} \sin \left ({\mathrm e}\right )}{24}-\frac {{\mathrm e}^{-3} \cos \left ({\mathrm e}\right ) {\mathrm e}}{12}\right ) \left (x -{\mathrm e}\right )^{4}+\left (-\frac {{\mathrm e}^{-4} {\mathrm e}^{2} \sin \left ({\mathrm e}\right )^{2}}{30}+\frac {\left (4 \cos \left ({\mathrm e}\right ) {\mathrm e}^{3}+3 \,{\mathrm e}^{2}-14\right ) {\mathrm e}^{-4} \sin \left ({\mathrm e}\right )}{120}+\frac {3 \cos \left ({\mathrm e}\right ) {\mathrm e}^{-4} \left ({\mathrm e}-\frac {{\mathrm e}^{3}}{9}\right )}{40}\right ) \left (x -{\mathrm e}\right )^{5}+\operatorname {O}\left (\left (x -{\mathrm e}\right )^{6}\right ) \]

23.7 problem 730

23.7.1 Existence and uniqueness analysis