problem 34.5 (d)

Internal problem ID [13947]
Internal ﬁle name [OUTPUT/13119_Friday_February_23_2024_06_55_06_AM_99309500/index.tex]

Book: Ordinary Diﬀerential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number: 34.5 (d).
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"

\[ \boxed {\sinh \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 2\).

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 \[ t = x -2 \] The ode is converted to be in terms of the new independent variable \(t\). This results in \[ \sinh \left (2+t \right ) \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+\left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right )-y \left (t \right ) {\mathrm e}^{2+t} = 0 \] With its expansion point and initial conditions now at \(t = 0\). 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 {-t^{2} \left (\frac {d}{d t}y \left (t \right )\right )+y \left (t \right ) {\mathrm e}^{2+t}-4 t \left (\frac {d}{d t}y \left (t \right )\right )-4 \frac {d}{d t}y \left (t \right )}{\sinh \left (2+t \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 \\ &= \left (\frac {\left (4+t \right ) \left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right ) {\mathrm e}^{-t -2}}{2}+\frac {\left (\frac {d}{d t}y \left (t \right )\right ) {\mathrm e}^{2 t +4}}{2}-\left (-\frac {t \left (\frac {d}{d t}y \left (t \right )\right )}{2}+y \left (t \right ) \left (2+t \right )\right ) \left (2+t \right ) {\mathrm e}^{2+t}+\left (t^{4}+8 t^{3}+24 t^{2}+32 t +\frac {31}{2}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-y \left (t \right )\right ) \operatorname {csch}\left (2+t \right )^{2}\\ 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 {4 \operatorname {csch}\left (2+t \right )^{2} \left (\left (\left (47+84 t +\frac {3}{2} t^{4}+15 t^{3}+54 t^{2}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-y \left (t \right )\right ) {\mathrm e}^{-t -2}+\left (\left (\frac {7}{2}+\frac {5}{4} t^{2}+4 t \right ) \left (\frac {d}{d t}y \left (t \right )\right )-y \left (t \right ) t \left (2+t \right )\right ) {\mathrm e}^{2 t +4}+\frac {\left (\frac {d}{d t}y \left (t \right )\right ) \left (t^{2}+8 t +14\right ) {\mathrm e}^{-2 t -4}}{4}-\frac {y \left (t \right ) {\mathrm e}^{3 t +6}}{2}+\left (\left (1+\frac {3}{2} t^{4}+9 t^{3}+18 t^{2}+12 t \right ) \left (\frac {d}{d t}y \left (t \right )\right )-\left (t^{4}+8 t^{3}+24 t^{2}+32 t +\frac {33}{2}\right ) y \left (t \right )\right ) {\mathrm e}^{2+t}+\left (65+12 t^{5}+60 t^{4}+\frac {481}{2} t^{2}+194 t +160 t^{3}+t^{6}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-2 y \left (t \right ) \left (3+t \right ) \left (2+t \right )\right )}{-2 \,{\mathrm e}^{-t -2}+2 \,{\mathrm e}^{2+t}}\\ 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 {4 \operatorname {csch}\left (2+t \right )^{2} \left (\left (\left (-\frac {839}{4}-t^{8}-16 t^{7}-1837 t^{2}-1016 t -112 t^{6}-1819 t^{3}-448 t^{5}-\frac {4497}{4} t^{4}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+\frac {7 y \left (t \right ) \left (t^{4}+\frac {66}{7} t^{3}+\frac {228}{7} t^{2}+\frac {344}{7} t +\frac {194}{7}\right )}{2}\right ) {\mathrm e}^{-t -2}+\left (\left (\frac {977}{4} t^{2}+116 t +120 t^{4}+240 t^{3}+3 t^{6}+30 t^{5}+\frac {43}{2}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-\left (t^{6}+12 t^{5}+60 t^{4}+160 t^{3}+\frac {977}{4} t^{2}+209 t +\frac {153}{2}\right ) y \left (t \right )\right ) {\mathrm e}^{2 t +4}+\left (\left (-3 t^{6}-42 t^{5}-\frac {4797}{4} t^{2}-1054 t -240 t^{4}-720 t^{3}-\frac {763}{2}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+\frac {13 \left (t^{2}+\frac {76}{13} t +\frac {106}{13}\right ) y \left (t \right )}{4}\right ) {\mathrm e}^{-2 t -4}+\left (\left (\frac {13}{4} t^{4}+40 t +41 t^{2}+19 t^{3}+\frac {67}{4}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-\frac {5 \left (t^{4}+6 t^{3}+12 t^{2}+8 t +\frac {6}{5}\right ) y \left (t \right )}{2}\right ) {\mathrm e}^{3 t +6}+\frac {\left (\left (-1+\frac {9}{2} t^{2}+7 t \right ) \left (\frac {d}{d t}y \left (t \right )\right )-7 y \left (t \right ) \left (t^{2}+\frac {16}{7} t +\frac {10}{7}\right )\right ) {\mathrm e}^{8+4 t}}{4}+\left (\left (-89 t^{2}-160 t -21 t^{3}-\frac {7}{4} t^{4}-\frac {205}{2}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+y \left (t \right )\right ) {\mathrm e}^{-3 t -6}-\frac {\left (\frac {d}{d t}y \left (t \right )\right ) \left (t^{2}+10 t +22\right ) {\mathrm e}^{-8-4 t}}{8}+\frac {\left (\frac {d}{d t}y \left (t \right )\right ) {\mathrm e}^{10+5 t}}{4}+\left (\left (\frac {4491}{4} t^{4}+t^{8}+16 t^{7}+112 t^{6}+448 t^{5}+1821 t^{3}+1885 t^{2}+1136 t +\frac {1181}{4}\right ) \left (\frac {d}{d t}y \left (t \right )\right )-y \left (t \right ) \left (t^{4}+18 t^{3}+84 t^{2}+152 t +95\right )\right ) {\mathrm e}^{2+t}+3 \left (121+\frac {625}{2} t +160 t^{3}+318 t^{2}+4 t^{5}+40 t^{4}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+y \left (t \right ) \left (\frac {105}{2}+160 t^{3}+\frac {971}{4} t^{2}+194 t +60 t^{4}+t^{6}+12 t^{5}\right )\right )}{\left ({\mathrm e}^{2+t}-{\mathrm e}^{-t -2}\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 \\ &= \text {Expression too large to display} \end {align*}

And so on. Evaluating all the above at initial conditions \(t = 0\) and \(y \left (0\right ) = y \left (0\right )\) and \(y^{\prime }\left (0\right ) = y^{\prime }\left (0\right )\) gives \begin {align*} F_0 &= \frac {2 \left (y \left (0\right ) {\mathrm e}^{2}-4 y^{\prime }\left (0\right )\right ) {\mathrm e}^{2}}{{\mathrm e}^{4}-1}\\ F_1 &= \frac {2 \,{\mathrm e}^{4} y^{\prime }\left (0\right )-16 y \left (0\right ) {\mathrm e}^{2}+16 y^{\prime }\left (0\right ) {\mathrm e}^{-2}-4 y \left (0\right )+62 y^{\prime }\left (0\right )}{{\mathrm e}^{4}-2+{\mathrm e}^{-4}}\\ F_2 &= \frac {28 \,{\mathrm e}^{4} y^{\prime }\left (0\right )-132 y \left (0\right ) {\mathrm e}^{2}+8 \,{\mathrm e}^{2} y^{\prime }\left (0\right )-8 y \left (0\right ) {\mathrm e}^{-2}+376 y^{\prime }\left (0\right ) {\mathrm e}^{-2}-4 y \left (0\right ) {\mathrm e}^{6}+28 y^{\prime }\left (0\right ) {\mathrm e}^{-4}-96 y \left (0\right )+520 y^{\prime }\left (0\right )}{-{\mathrm e}^{6}+3 \,{\mathrm e}^{2}-3 \,{\mathrm e}^{-2}+{\mathrm e}^{-6}}\\ F_3 &= \frac {-40 y \left (0\right ) {\mathrm e}^{8}-4 y^{\prime }\left (0\right ) {\mathrm e}^{8}-44 y^{\prime }\left (0\right ) {\mathrm e}^{-8}+4 y^{\prime }\left (0\right ) {\mathrm e}^{10}-1224 y \left (0\right ) {\mathrm e}^{4}+344 \,{\mathrm e}^{4} y^{\prime }\left (0\right )-1520 y \left (0\right ) {\mathrm e}^{2}+4724 \,{\mathrm e}^{2} y^{\prime }\left (0\right )+1552 y \left (0\right ) {\mathrm e}^{-2}-3356 y^{\prime }\left (0\right ) {\mathrm e}^{-2}+424 y \left (0\right ) {\mathrm e}^{-4}-6104 y^{\prime }\left (0\right ) {\mathrm e}^{-4}-48 y \left (0\right ) {\mathrm e}^{6}+268 y^{\prime }\left (0\right ) {\mathrm e}^{6}+16 y \left (0\right ) {\mathrm e}^{-6}-1640 y^{\prime }\left (0\right ) {\mathrm e}^{-6}+840 y \left (0\right )+5808 y^{\prime }\left (0\right )}{{\mathrm e}^{10}-5 \,{\mathrm e}^{6}+10 \,{\mathrm e}^{2}-10 \,{\mathrm e}^{-2}+5 \,{\mathrm e}^{-6}-{\mathrm e}^{-10}}\\ F_4 &= \frac {-10648 y \left (0\right ) {\mathrm e}^{-2}+39992 y^{\prime }\left (0\right ) {\mathrm e}^{-4}+20992 y \left (0\right ) {\mathrm e}^{4}-44608 y \left (0\right )+62288 y^{\prime }\left (0\right )+173568 y^{\prime }\left (0\right ) {\mathrm e}^{-2}+13488 y \left (0\right ) {\mathrm e}^{6}-32 y \left (0\right ) {\mathrm e}^{12}-78304 \,{\mathrm e}^{2} y^{\prime }\left (0\right )-15520 y \left (0\right ) {\mathrm e}^{2}-51536 \,{\mathrm e}^{4} y^{\prime }\left (0\right )+1472 y \left (0\right ) {\mathrm e}^{8}+20512 y \left (0\right ) {\mathrm e}^{-4}+12336 y \left (0\right ) {\mathrm e}^{-6}-2456 y^{\prime }\left (0\right ) {\mathrm e}^{8}-48184 y^{\prime }\left (0\right ) {\mathrm e}^{-8}+48 y^{\prime }\left (0\right ) {\mathrm e}^{10}-7776 y^{\prime }\left (0\right ) {\mathrm e}^{6}-81360 y^{\prime }\left (0\right ) {\mathrm e}^{-6}+32 y \left (0\right ) {\mathrm e}^{-10}+8 y \left (0\right ) {\mathrm e}^{14}+304 y \left (0\right ) {\mathrm e}^{10}+1664 y \left (0\right ) {\mathrm e}^{-8}-64 y^{\prime }\left (0\right ) {\mathrm e}^{-12}-40 y^{\prime }\left (0\right ) {\mathrm e}^{12}-6176 y^{\prime }\left (0\right ) {\mathrm e}^{-10}}{{\mathrm e}^{14}-7 \,{\mathrm e}^{10}+21 \,{\mathrm e}^{6}-35 \,{\mathrm e}^{2}+35 \,{\mathrm e}^{-2}-21 \,{\mathrm e}^{-6}+7 \,{\mathrm e}^{-10}-{\mathrm e}^{-14}} \end {align*}

Substituting all the above in (7) and simplifying gives the solution as \[ \text {Expression too large to display} \] 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 {-t^{2} \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right ) {\mathrm e}^{2+t}-4 t \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\right )-4 \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\right )}{\sinh \left (2+t \right )}\tag {1} \end {align*}

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

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

Hence the ODE in Eq (1) becomes \[ \left (\sinh \left (2\right )+\cosh \left (2\right ) t +\frac {\sinh \left (2\right ) t^{2}}{2}+\frac {\cosh \left (2\right ) t^{3}}{6}+\frac {\sinh \left (2\right ) t^{4}}{24}+\frac {\cosh \left (2\right ) t^{5}}{120}+\frac {\sinh \left (2\right ) t^{6}}{720}\right ) \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\left (t^{2}+4 t +4\right ) \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\right )+\left (-{\mathrm e}^{2}-{\mathrm e}^{2} t -\frac {{\mathrm e}^{2} t^{2}}{2}-\frac {{\mathrm e}^{2} t^{3}}{6}-\frac {t^{4} {\mathrm e}^{2}}{24}-\frac {t^{5} {\mathrm e}^{2}}{120}-\frac {t^{6} {\mathrm e}^{2}}{720}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right ) = 0 \] Expanding the ﬁrst term in (1) gives \[ \sinh \left (2\right )\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\cosh \left (2\right ) t \cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\sinh \left (2\right ) t^{2}}{2}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\cosh \left (2\right ) t^{3}}{6}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\sinh \left (2\right ) t^{4}}{24}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\cosh \left (2\right ) t^{5}}{120}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\sinh \left (2\right ) t^{6}}{720}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\left (t^{2}+4 t +4\right ) \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\right )+\left (-{\mathrm e}^{2}-{\mathrm e}^{2} t -\frac {{\mathrm e}^{2} t^{2}}{2}-\frac {{\mathrm e}^{2} t^{3}}{6}-\frac {t^{4} {\mathrm e}^{2}}{24}-\frac {t^{5} {\mathrm e}^{2}}{120}-\frac {t^{6} {\mathrm e}^{2}}{720}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right ) = 0 \] Expanding the third term in (1) gives \[ \text {Expression too large to display} \] Which simpliﬁes to \begin{equation} \tag{2} \left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +4} a_{n} \left (n -1\right ) \sinh \left (2\right )}{720}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +3} a_{n} \left (n -1\right ) \cosh \left (2\right )}{120}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +2} a_{n} \left (n -1\right ) \sinh \left (2\right )}{24}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{1+n} a_{n} \left (n -1\right ) \cosh \left (2\right )}{6}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {n a_{n} t^{n} \sinh \left (2\right ) \left (n -1\right )}{2}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}n \,t^{n -1} a_{n} \left (n -1\right ) \cosh \left (2\right )\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}n \,t^{n -2} a_{n} \left (n -1\right ) \sinh \left (2\right )\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}n \,t^{1+n} a_{n}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}4 n a_{n} t^{n}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}4 n a_{n} t^{n -1}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-a_{n} t^{n} {\mathrm e}^{2}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-t^{1+n} a_{n} {\mathrm e}^{2}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +2} a_{n} {\mathrm e}^{2}}{2}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +3} a_{n} {\mathrm e}^{2}}{6}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +4} a_{n} {\mathrm e}^{2}}{24}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +5} a_{n} {\mathrm e}^{2}}{120}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +6} a_{n} {\mathrm e}^{2}}{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 }}\frac {n \,t^{n +4} a_{n} \left (n -1\right ) \sinh \left (2\right )}{720} &= \moverset {\infty }{\munderset {n =6}{\sum }}\frac {\left (n -4\right ) a_{n -4} \left (n -5\right ) \sinh \left (2\right ) t^{n}}{720} \\ \moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +3} a_{n} \left (n -1\right ) \cosh \left (2\right )}{120} &= \moverset {\infty }{\munderset {n =5}{\sum }}\frac {\left (n -3\right ) a_{n -3} \left (n -4\right ) \cosh \left (2\right ) t^{n}}{120} \\ \moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +2} a_{n} \left (n -1\right ) \sinh \left (2\right )}{24} &= \moverset {\infty }{\munderset {n =4}{\sum }}\frac {\left (n -2\right ) a_{n -2} \left (n -3\right ) \sinh \left (2\right ) t^{n}}{24} \\ \moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{1+n} a_{n} \left (n -1\right ) \cosh \left (2\right )}{6} &= \moverset {\infty }{\munderset {n =3}{\sum }}\frac {\left (n -1\right ) a_{n -1} \left (n -2\right ) \cosh \left (2\right ) t^{n}}{6} \\ \moverset {\infty }{\munderset {n =2}{\sum }}n \,t^{n -1} a_{n} \left (n -1\right ) \cosh \left (2\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (1+n \right ) a_{1+n} n \cosh \left (2\right ) t^{n} \\ \moverset {\infty }{\munderset {n =2}{\sum }}n \,t^{n -2} a_{n} \left (n -1\right ) \sinh \left (2\right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +2\right ) a_{n +2} \left (1+n \right ) \sinh \left (2\right ) t^{n} \\ \moverset {\infty }{\munderset {n =1}{\sum }}n \,t^{1+n} a_{n} &= \moverset {\infty }{\munderset {n =2}{\sum }}\left (n -1\right ) a_{n -1} t^{n} \\ \moverset {\infty }{\munderset {n =1}{\sum }}4 n a_{n} t^{n -1} &= \moverset {\infty }{\munderset {n =0}{\sum }}4 \left (1+n \right ) a_{1+n} t^{n} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-t^{1+n} a_{n} {\mathrm e}^{2}\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} {\mathrm e}^{2} t^{n}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +2} a_{n} {\mathrm e}^{2}}{2}\right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {a_{n -2} {\mathrm e}^{2} t^{n}}{2}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +3} a_{n} {\mathrm e}^{2}}{6}\right ) &= \moverset {\infty }{\munderset {n =3}{\sum }}\left (-\frac {a_{n -3} {\mathrm e}^{2} t^{n}}{6}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +4} a_{n} {\mathrm e}^{2}}{24}\right ) &= \moverset {\infty }{\munderset {n =4}{\sum }}\left (-\frac {a_{n -4} {\mathrm e}^{2} t^{n}}{24}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +5} a_{n} {\mathrm e}^{2}}{120}\right ) &= \moverset {\infty }{\munderset {n =5}{\sum }}\left (-\frac {a_{n -5} {\mathrm e}^{2} t^{n}}{120}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +6} a_{n} {\mathrm e}^{2}}{720}\right ) &= \moverset {\infty }{\munderset {n =6}{\sum }}\left (-\frac {a_{n -6} {\mathrm e}^{2} t^{n}}{720}\right ) \\ \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} \left (\moverset {\infty }{\munderset {n =6}{\sum }}\frac {\left (n -4\right ) a_{n -4} \left (n -5\right ) \sinh \left (2\right ) t^{n}}{720}\right )+\left (\moverset {\infty }{\munderset {n =5}{\sum }}\frac {\left (n -3\right ) a_{n -3} \left (n -4\right ) \cosh \left (2\right ) t^{n}}{120}\right )+\left (\moverset {\infty }{\munderset {n =4}{\sum }}\frac {\left (n -2\right ) a_{n -2} \left (n -3\right ) \sinh \left (2\right ) t^{n}}{24}\right )+\left (\moverset {\infty }{\munderset {n =3}{\sum }}\frac {\left (n -1\right ) a_{n -1} \left (n -2\right ) \cosh \left (2\right ) t^{n}}{6}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {n a_{n} t^{n} \sinh \left (2\right ) \left (n -1\right )}{2}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}\left (1+n \right ) a_{1+n} n \cosh \left (2\right ) t^{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +2\right ) a_{n +2} \left (1+n \right ) \sinh \left (2\right ) t^{n}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\left (n -1\right ) a_{n -1} t^{n}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}4 n a_{n} t^{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 \left (1+n \right ) a_{1+n} t^{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-a_{n} t^{n} {\mathrm e}^{2}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} {\mathrm e}^{2} t^{n}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {a_{n -2} {\mathrm e}^{2} t^{n}}{2}\right )+\moverset {\infty }{\munderset {n =3}{\sum }}\left (-\frac {a_{n -3} {\mathrm e}^{2} t^{n}}{6}\right )+\moverset {\infty }{\munderset {n =4}{\sum }}\left (-\frac {a_{n -4} {\mathrm e}^{2} t^{n}}{24}\right )+\moverset {\infty }{\munderset {n =5}{\sum }}\left (-\frac {a_{n -5} {\mathrm e}^{2} t^{n}}{120}\right )+\moverset {\infty }{\munderset {n =6}{\sum }}\left (-\frac {a_{n -6} {\mathrm e}^{2} t^{n}}{720}\right ) = 0 \end{equation} \(n=0\) gives \[ 2 a_{2} \sinh \left (2\right )+4 a_{1}-a_{0} {\mathrm e}^{2}=0 \] \[ a_{2} = \frac {a_{0} {\mathrm e}^{2}-4 a_{1}}{2 \sinh \left (2\right )} \] \(n=1\) gives \[ 2 a_{2} \cosh \left (2\right )+6 a_{3} \sinh \left (2\right )+4 a_{1}+8 a_{2}-a_{1} {\mathrm e}^{2}-a_{0} {\mathrm e}^{2}=0 \] Which after substituting earlier equations, simpliﬁes to \[ a_{3} = \frac {a_{0} {\mathrm e}^{2} \sinh \left (2\right )+a_{1} {\mathrm e}^{2} \sinh \left (2\right )-\cosh \left (2\right ) a_{0} {\mathrm e}^{2}-4 a_{0} {\mathrm e}^{2}-4 a_{1} \sinh \left (2\right )+4 \cosh \left (2\right ) a_{1}+16 a_{1}}{6 \sinh \left (2\right )^{2}} \] \(n=2\) gives \[ a_{2} \sinh \left (2\right )+6 a_{3} \cosh \left (2\right )+12 a_{4} \sinh \left (2\right )+a_{1}+8 a_{2}+12 a_{3}-a_{2} {\mathrm e}^{2}-a_{1} {\mathrm e}^{2}-\frac {a_{0} {\mathrm e}^{2}}{2}=0 \] Which after substituting earlier equations, simpliﬁes to \[ a_{4} = \frac {{\mathrm e}^{4} a_{0} \sinh \left (2\right )+2 a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{2}-2 \cosh \left (2\right ) a_{0} {\mathrm e}^{2} \sinh \left (2\right )-2 \cosh \left (2\right ) a_{1} {\mathrm e}^{2} \sinh \left (2\right )+2 \cosh \left (2\right )^{2} a_{0} {\mathrm e}^{2}-12 a_{0} {\mathrm e}^{2} \sinh \left (2\right )-8 a_{1} {\mathrm e}^{2} \sinh \left (2\right )+12 \cosh \left (2\right ) a_{0} {\mathrm e}^{2}+2 a_{1} \sinh \left (2\right )^{2}+8 \cosh \left (2\right ) a_{1} \sinh \left (2\right )-8 \cosh \left (2\right )^{2} a_{1}+16 a_{0} {\mathrm e}^{2}+48 a_{1} \sinh \left (2\right )-48 \cosh \left (2\right ) a_{1}-64 a_{1}}{24 \sinh \left (2\right )^{3}} \] \(n=3\) gives \[ \frac {a_{2} \cosh \left (2\right )}{3}+3 a_{3} \sinh \left (2\right )+12 a_{4} \cosh \left (2\right )+20 a_{5} \sinh \left (2\right )+2 a_{2}+12 a_{3}+16 a_{4}-a_{3} {\mathrm e}^{2}-a_{2} {\mathrm e}^{2}-\frac {a_{1} {\mathrm e}^{2}}{2}-\frac {a_{0} {\mathrm e}^{2}}{6}=0 \] Which after substituting earlier equations, simpliﬁes to \[ a_{5} = \frac {-2 a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{3}-6 \cosh \left (2\right )^{3} a_{0} {\mathrm e}^{2}-6 a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{2}-14 \cosh \left (2\right ) a_{1} \sinh \left (2\right )^{2}-24 \cosh \left (2\right )^{2} a_{1} \sinh \left (2\right )+4 \,{\mathrm e}^{4} a_{0} \sinh \left (2\right )^{2}+{\mathrm e}^{4} a_{1} \sinh \left (2\right )^{2}+96 a_{0} {\mathrm e}^{2} \sinh \left (2\right )+48 a_{1} {\mathrm e}^{2} \sinh \left (2\right )-96 \cosh \left (2\right ) a_{0} {\mathrm e}^{2}+16 a_{1} \sinh \left (2\right )^{2}+176 \cosh \left (2\right )^{2} a_{1}-384 a_{1} \sinh \left (2\right )+384 \cosh \left (2\right ) a_{1}-36 a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{2}-44 \cosh \left (2\right )^{2} a_{0} {\mathrm e}^{2}-224 \cosh \left (2\right ) a_{1} \sinh \left (2\right )-8 \,{\mathrm e}^{4} a_{0} \sinh \left (2\right )+256 a_{1}+12 a_{1} \sinh \left (2\right )^{3}+24 \cosh \left (2\right )^{3} a_{1}+56 \cosh \left (2\right ) a_{0} {\mathrm e}^{2} \sinh \left (2\right )+36 \cosh \left (2\right ) a_{1} {\mathrm e}^{2} \sinh \left (2\right )-64 a_{0} {\mathrm e}^{2}+2 \cosh \left (2\right ) a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{2}-6 \cosh \left (2\right ) a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{2}+6 \cosh \left (2\right )^{2} a_{0} {\mathrm e}^{2} \sinh \left (2\right )+6 \cosh \left (2\right )^{2} a_{1} {\mathrm e}^{2} \sinh \left (2\right )-4 \cosh \left (2\right ) {\mathrm e}^{4} a_{0} \sinh \left (2\right )}{120 \sinh \left (2\right )^{4}} \] \(n=4\) gives \[ \frac {a_{2} \sinh \left (2\right )}{12}+a_{3} \cosh \left (2\right )+6 a_{4} \sinh \left (2\right )+20 a_{5} \cosh \left (2\right )+30 a_{6} \sinh \left (2\right )+3 a_{3}+16 a_{4}+20 a_{5}-a_{4} {\mathrm e}^{2}-a_{3} {\mathrm e}^{2}-\frac {a_{2} {\mathrm e}^{2}}{2}-\frac {a_{1} {\mathrm e}^{2}}{6}-\frac {a_{0} {\mathrm e}^{2}}{24}=0 \] Which after substituting earlier equations, simpliﬁes to \[ a_{6} = \frac {16 \cosh \left (2\right ) a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{3}+8 \cosh \left (2\right ) a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{3}-16 \cosh \left (2\right )^{2} a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{2}+24 \cosh \left (2\right )^{2} a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{2}-24 \cosh \left (2\right )^{3} a_{0} {\mathrm e}^{2} \sinh \left (2\right )-24 \cosh \left (2\right )^{3} a_{1} {\mathrm e}^{2} \sinh \left (2\right )-22 \cosh \left (2\right ) {\mathrm e}^{4} a_{0} \sinh \left (2\right )^{2}-6 \cosh \left (2\right ) {\mathrm e}^{4} a_{1} \sinh \left (2\right )^{2}+18 \cosh \left (2\right )^{2} {\mathrm e}^{4} a_{0} \sinh \left (2\right )+68 a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{3}+200 \cosh \left (2\right )^{3} a_{0} {\mathrm e}^{2}+168 a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{2}+40 \cosh \left (2\right ) a_{1} \sinh \left (2\right )^{2}+1120 \cosh \left (2\right )^{2} a_{1} \sinh \left (2\right )-60 \,{\mathrm e}^{4} a_{0} \sinh \left (2\right )^{2}-12 \,{\mathrm e}^{4} a_{1} \sinh \left (2\right )^{2}-640 a_{0} {\mathrm e}^{2} \sinh \left (2\right )-256 a_{1} {\mathrm e}^{2} \sinh \left (2\right )+640 \cosh \left (2\right ) a_{0} {\mathrm e}^{2}-640 a_{1} \sinh \left (2\right )^{2}-2240 \cosh \left (2\right )^{2} a_{1}+2560 a_{1} \sinh \left (2\right )-2560 \cosh \left (2\right ) a_{1}+384 a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{2}+560 \cosh \left (2\right )^{2} a_{0} {\mathrm e}^{2}+3200 \cosh \left (2\right ) a_{1} \sinh \left (2\right )+48 \,{\mathrm e}^{4} a_{0} \sinh \left (2\right )-1024 a_{1}-8 a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{4}+24 \cosh \left (2\right )^{4} a_{0} {\mathrm e}^{2}-34 a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{3}-80 \cosh \left (2\right ) a_{1} \sinh \left (2\right )^{3}+88 \cosh \left (2\right )^{2} a_{1} \sinh \left (2\right )^{2}+96 \cosh \left (2\right )^{3} a_{1} \sinh \left (2\right )+{\mathrm e}^{6} a_{0} \sinh \left (2\right )^{2}+4 \,{\mathrm e}^{4} a_{0} \sinh \left (2\right )^{3}+6 \,{\mathrm e}^{4} a_{1} \sinh \left (2\right )^{3}-8 a_{1} \sinh \left (2\right )^{4}-96 \cosh \left (2\right )^{4} a_{1}-320 a_{1} \sinh \left (2\right )^{3}-800 \cosh \left (2\right )^{3} a_{1}-800 \cosh \left (2\right ) a_{0} {\mathrm e}^{2} \sinh \left (2\right )-384 \cosh \left (2\right ) a_{1} {\mathrm e}^{2} \sinh \left (2\right )+256 a_{0} {\mathrm e}^{2}+4 \cosh \left (2\right ) a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{2}+224 \cosh \left (2\right ) a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{2}-280 \cosh \left (2\right )^{2} a_{0} {\mathrm e}^{2} \sinh \left (2\right )-176 \cosh \left (2\right )^{2} a_{1} {\mathrm e}^{2} \sinh \left (2\right )+60 \cosh \left (2\right ) {\mathrm e}^{4} a_{0} \sinh \left (2\right )}{720 \sinh \left (2\right )^{5}} \] \(n=5\) gives \[ \frac {a_{2} \cosh \left (2\right )}{60}+\frac {a_{3} \sinh \left (2\right )}{4}+2 a_{4} \cosh \left (2\right )+10 a_{5} \sinh \left (2\right )+30 a_{6} \cosh \left (2\right )+42 a_{7} \sinh \left (2\right )+4 a_{4}+20 a_{5}+24 a_{6}-a_{5} {\mathrm e}^{2}-a_{4} {\mathrm e}^{2}-\frac {a_{3} {\mathrm e}^{2}}{2}-\frac {a_{2} {\mathrm e}^{2}}{6}-\frac {a_{1} {\mathrm e}^{2}}{24}-\frac {a_{0} {\mathrm e}^{2}}{120}=0 \] Which after substituting earlier equations, simpliﬁes to \[ \text {Expression too large to display} \] For \(6\le n\), the recurrence equation is \begin{equation} \tag{4} \frac {\left (n -4\right ) a_{n -4} \left (n -5\right ) \sinh \left (2\right )}{720}+\frac {\left (n -3\right ) a_{n -3} \left (n -4\right ) \cosh \left (2\right )}{120}+\frac {\left (n -2\right ) a_{n -2} \left (n -3\right ) \sinh \left (2\right )}{24}+\frac {\left (n -1\right ) a_{n -1} \left (n -2\right ) \cosh \left (2\right )}{6}+\frac {n a_{n} \left (n -1\right ) \sinh \left (2\right )}{2}+\left (1+n \right ) a_{1+n} n \cosh \left (2\right )+\left (n +2\right ) a_{n +2} \left (1+n \right ) \sinh \left (2\right )+\left (n -1\right ) a_{n -1}+4 n a_{n}+4 \left (1+n \right ) a_{1+n}-a_{n} {\mathrm e}^{2}-a_{n -1} {\mathrm e}^{2}-\frac {a_{n -2} {\mathrm e}^{2}}{2}-\frac {a_{n -3} {\mathrm e}^{2}}{6}-\frac {a_{n -4} {\mathrm e}^{2}}{24}-\frac {a_{n -5} {\mathrm e}^{2}}{120}-\frac {a_{n -6} {\mathrm e}^{2}}{720} = 0 \end{equation} Solving for \(a_{n +2}\), gives \begin{align*} \tag{5} a_{n +2}&= \frac {720 a_{n -1}-720 a_{n -1} n -2880 a_{1+n} n -20 a_{n -4} \sinh \left (2\right )-180 a_{n -2} \sinh \left (2\right )-72 a_{n -3} \cosh \left (2\right )-240 a_{n -1} \cosh \left (2\right )-360 n^{2} a_{n} \sinh \left (2\right )-\sinh \left (2\right ) n^{2} a_{n -4}-30 \sinh \left (2\right ) n^{2} a_{n -2}-720 a_{1+n} n^{2} \cosh \left (2\right )-6 \cosh \left (2\right ) n^{2} a_{n -3}-120 \cosh \left (2\right ) n^{2} a_{n -1}+360 n a_{n} \sinh \left (2\right )+9 \sinh \left (2\right ) n a_{n -4}+150 \sinh \left (2\right ) n a_{n -2}-720 a_{1+n} n \cosh \left (2\right )+42 \cosh \left (2\right ) n a_{n -3}+360 \cosh \left (2\right ) n a_{n -1}+360 a_{n -2} {\mathrm e}^{2}+120 a_{n -3} {\mathrm e}^{2}-2880 n a_{n}+720 a_{n} {\mathrm e}^{2}+30 a_{n -4} {\mathrm e}^{2}+6 a_{n -5} {\mathrm e}^{2}-2880 a_{1+n}+720 a_{n -1} {\mathrm e}^{2}+a_{n -6} {\mathrm e}^{2}}{720 \sinh \left (2\right ) \left (n^{2}+3 n +2\right )} \\ &= \frac {\left (-360 \sinh \left (2\right ) n^{2}+360 \sinh \left (2\right ) n +720 \,{\mathrm e}^{2}-2880 n \right ) a_{n}}{720 \sinh \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\left (-720 \cosh \left (2\right ) n^{2}-720 \cosh \left (2\right ) n -2880 n -2880\right ) a_{1+n}}{720 \sinh \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {{\mathrm e}^{2} a_{n -6}}{720 \sinh \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {{\mathrm e}^{2} a_{n -5}}{120 \sinh \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\left (-\sinh \left (2\right ) n^{2}+9 \sinh \left (2\right ) n +30 \,{\mathrm e}^{2}-20 \sinh \left (2\right )\right ) a_{n -4}}{720 \sinh \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\left (-6 \cosh \left (2\right ) n^{2}+42 \cosh \left (2\right ) n +120 \,{\mathrm e}^{2}-72 \cosh \left (2\right )\right ) a_{n -3}}{720 \sinh \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\left (-30 \sinh \left (2\right ) n^{2}+150 \sinh \left (2\right ) n +360 \,{\mathrm e}^{2}-180 \sinh \left (2\right )\right ) a_{n -2}}{720 \sinh \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\left (-120 \cosh \left (2\right ) n^{2}+360 \cosh \left (2\right ) n +720 \,{\mathrm e}^{2}-240 \cosh \left (2\right )-720 n +720\right ) a_{n -1}}{720 \sinh \left (2\right ) \left (n^{2}+3 n +2\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 {\left (a_{0} {\mathrm e}^{2}-4 a_{1}\right ) t^{2}}{2 \sinh \left (2\right )}+\frac {\left (a_{0} {\mathrm e}^{2} \sinh \left (2\right )+a_{1} {\mathrm e}^{2} \sinh \left (2\right )-\cosh \left (2\right ) a_{0} {\mathrm e}^{2}-4 a_{0} {\mathrm e}^{2}-4 a_{1} \sinh \left (2\right )+4 \cosh \left (2\right ) a_{1}+16 a_{1}\right ) t^{3}}{6 \sinh \left (2\right )^{2}}+\frac {\left ({\mathrm e}^{4} a_{0} \sinh \left (2\right )+2 a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{2}-2 \cosh \left (2\right ) a_{0} {\mathrm e}^{2} \sinh \left (2\right )-2 \cosh \left (2\right ) a_{1} {\mathrm e}^{2} \sinh \left (2\right )+2 \cosh \left (2\right )^{2} a_{0} {\mathrm e}^{2}-12 a_{0} {\mathrm e}^{2} \sinh \left (2\right )-8 a_{1} {\mathrm e}^{2} \sinh \left (2\right )+12 \cosh \left (2\right ) a_{0} {\mathrm e}^{2}+2 a_{1} \sinh \left (2\right )^{2}+8 \cosh \left (2\right ) a_{1} \sinh \left (2\right )-8 \cosh \left (2\right )^{2} a_{1}+16 a_{0} {\mathrm e}^{2}+48 a_{1} \sinh \left (2\right )-48 \cosh \left (2\right ) a_{1}-64 a_{1}\right ) t^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left (-2 a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{3}-6 \cosh \left (2\right )^{3} a_{0} {\mathrm e}^{2}-6 a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{2}-14 \cosh \left (2\right ) a_{1} \sinh \left (2\right )^{2}-24 \cosh \left (2\right )^{2} a_{1} \sinh \left (2\right )+4 \,{\mathrm e}^{4} a_{0} \sinh \left (2\right )^{2}+{\mathrm e}^{4} a_{1} \sinh \left (2\right )^{2}+96 a_{0} {\mathrm e}^{2} \sinh \left (2\right )+48 a_{1} {\mathrm e}^{2} \sinh \left (2\right )-96 \cosh \left (2\right ) a_{0} {\mathrm e}^{2}+16 a_{1} \sinh \left (2\right )^{2}+176 \cosh \left (2\right )^{2} a_{1}-384 a_{1} \sinh \left (2\right )+384 \cosh \left (2\right ) a_{1}-36 a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{2}-44 \cosh \left (2\right )^{2} a_{0} {\mathrm e}^{2}-224 \cosh \left (2\right ) a_{1} \sinh \left (2\right )-8 \,{\mathrm e}^{4} a_{0} \sinh \left (2\right )+256 a_{1}+12 a_{1} \sinh \left (2\right )^{3}+24 \cosh \left (2\right )^{3} a_{1}+56 \cosh \left (2\right ) a_{0} {\mathrm e}^{2} \sinh \left (2\right )+36 \cosh \left (2\right ) a_{1} {\mathrm e}^{2} \sinh \left (2\right )-64 a_{0} {\mathrm e}^{2}+2 \cosh \left (2\right ) a_{0} {\mathrm e}^{2} \sinh \left (2\right )^{2}-6 \cosh \left (2\right ) a_{1} {\mathrm e}^{2} \sinh \left (2\right )^{2}+6 \cosh \left (2\right )^{2} a_{0} {\mathrm e}^{2} \sinh \left (2\right )+6 \cosh \left (2\right )^{2} a_{1} {\mathrm e}^{2} \sinh \left (2\right )-4 \cosh \left (2\right ) {\mathrm e}^{4} a_{0} \sinh \left (2\right )\right ) t^{5}}{120 \sinh \left (2\right )^{4}}+\dots \] Collecting terms, the solution becomes \begin{equation} \tag{3} y \left (t \right ) = \left (1+\frac {{\mathrm e}^{2} t^{2}}{2 \sinh \left (2\right )}+\frac {\left ({\mathrm e}^{2} \sinh \left (2\right )-{\mathrm e}^{2} \cosh \left (2\right )-4 \,{\mathrm e}^{2}\right ) t^{3}}{6 \sinh \left (2\right )^{2}}+\frac {\left ({\mathrm e}^{4} \sinh \left (2\right )-2 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )+2 \,{\mathrm e}^{2} \cosh \left (2\right )^{2}-12 \,{\mathrm e}^{2} \sinh \left (2\right )+12 \,{\mathrm e}^{2} \cosh \left (2\right )+16 \,{\mathrm e}^{2}\right ) t^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left (4 \,{\mathrm e}^{4} \sinh \left (2\right )^{2}-4 \,{\mathrm e}^{4} \sinh \left (2\right ) \cosh \left (2\right )-2 \,{\mathrm e}^{2} \sinh \left (2\right )^{3}+2 \,{\mathrm e}^{2} \sinh \left (2\right )^{2} \cosh \left (2\right )+6 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )^{2}-6 \,{\mathrm e}^{2} \cosh \left (2\right )^{3}-8 \,{\mathrm e}^{4} \sinh \left (2\right )-6 \,{\mathrm e}^{2} \sinh \left (2\right )^{2}+56 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )-44 \,{\mathrm e}^{2} \cosh \left (2\right )^{2}+96 \,{\mathrm e}^{2} \sinh \left (2\right )-96 \,{\mathrm e}^{2} \cosh \left (2\right )-64 \,{\mathrm e}^{2}\right ) t^{5}}{120 \sinh \left (2\right )^{4}}\right ) a_{0}+\left (t -\frac {2 t^{2}}{\sinh \left (2\right )}+\frac {\left ({\mathrm e}^{2} \sinh \left (2\right )-4 \sinh \left (2\right )+4 \cosh \left (2\right )+16\right ) t^{3}}{6 \sinh \left (2\right )^{2}}+\frac {\left (2 \,{\mathrm e}^{2} \sinh \left (2\right )^{2}-2 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )-8 \,{\mathrm e}^{2} \sinh \left (2\right )+2 \sinh \left (2\right )^{2}+8 \sinh \left (2\right ) \cosh \left (2\right )-8 \cosh \left (2\right )^{2}+48 \sinh \left (2\right )-48 \cosh \left (2\right )-64\right ) t^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left ({\mathrm e}^{4} \sinh \left (2\right )^{2}-6 \,{\mathrm e}^{2} \sinh \left (2\right )^{2} \cosh \left (2\right )+6 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )^{2}-36 \,{\mathrm e}^{2} \sinh \left (2\right )^{2}+36 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )+12 \sinh \left (2\right )^{3}-14 \sinh \left (2\right )^{2} \cosh \left (2\right )-24 \sinh \left (2\right ) \cosh \left (2\right )^{2}+24 \cosh \left (2\right )^{3}+48 \,{\mathrm e}^{2} \sinh \left (2\right )+16 \sinh \left (2\right )^{2}-224 \sinh \left (2\right ) \cosh \left (2\right )+176 \cosh \left (2\right )^{2}-384 \sinh \left (2\right )+384 \cosh \left (2\right )+256\right ) t^{5}}{120 \sinh \left (2\right )^{4}}\right ) a_{1}+O\left (t^{6}\right ) \end{equation} At \(t = 0\) the solution above becomes \[ y \left (t \right ) = \left (1+\frac {{\mathrm e}^{2} t^{2}}{2 \sinh \left (2\right )}+\frac {\left ({\mathrm e}^{2} \sinh \left (2\right )-{\mathrm e}^{2} \cosh \left (2\right )-4 \,{\mathrm e}^{2}\right ) t^{3}}{6 \sinh \left (2\right )^{2}}+\frac {\left ({\mathrm e}^{4} \sinh \left (2\right )-2 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )+2 \,{\mathrm e}^{2} \cosh \left (2\right )^{2}-12 \,{\mathrm e}^{2} \sinh \left (2\right )+12 \,{\mathrm e}^{2} \cosh \left (2\right )+16 \,{\mathrm e}^{2}\right ) t^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left (4 \,{\mathrm e}^{4} \sinh \left (2\right )^{2}-4 \,{\mathrm e}^{4} \sinh \left (2\right ) \cosh \left (2\right )-2 \,{\mathrm e}^{2} \sinh \left (2\right )^{3}+2 \,{\mathrm e}^{2} \sinh \left (2\right )^{2} \cosh \left (2\right )+6 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )^{2}-6 \,{\mathrm e}^{2} \cosh \left (2\right )^{3}-8 \,{\mathrm e}^{4} \sinh \left (2\right )-6 \,{\mathrm e}^{2} \sinh \left (2\right )^{2}+56 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )-44 \,{\mathrm e}^{2} \cosh \left (2\right )^{2}+96 \,{\mathrm e}^{2} \sinh \left (2\right )-96 \,{\mathrm e}^{2} \cosh \left (2\right )-64 \,{\mathrm e}^{2}\right ) t^{5}}{120 \sinh \left (2\right )^{4}}\right ) c_{1} +\left (t -\frac {2 t^{2}}{\sinh \left (2\right )}+\frac {\left ({\mathrm e}^{2} \sinh \left (2\right )-4 \sinh \left (2\right )+4 \cosh \left (2\right )+16\right ) t^{3}}{6 \sinh \left (2\right )^{2}}+\frac {\left (2 \,{\mathrm e}^{2} \sinh \left (2\right )^{2}-2 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )-8 \,{\mathrm e}^{2} \sinh \left (2\right )+2 \sinh \left (2\right )^{2}+8 \sinh \left (2\right ) \cosh \left (2\right )-8 \cosh \left (2\right )^{2}+48 \sinh \left (2\right )-48 \cosh \left (2\right )-64\right ) t^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left ({\mathrm e}^{4} \sinh \left (2\right )^{2}-6 \,{\mathrm e}^{2} \sinh \left (2\right )^{2} \cosh \left (2\right )+6 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )^{2}-36 \,{\mathrm e}^{2} \sinh \left (2\right )^{2}+36 \,{\mathrm e}^{2} \sinh \left (2\right ) \cosh \left (2\right )+12 \sinh \left (2\right )^{3}-14 \sinh \left (2\right )^{2} \cosh \left (2\right )-24 \sinh \left (2\right ) \cosh \left (2\right )^{2}+24 \cosh \left (2\right )^{3}+48 \,{\mathrm e}^{2} \sinh \left (2\right )+16 \sinh \left (2\right )^{2}-224 \sinh \left (2\right ) \cosh \left (2\right )+176 \cosh \left (2\right )^{2}-384 \sinh \left (2\right )+384 \cosh \left (2\right )+256\right ) t^{5}}{120 \sinh \left (2\right )^{4}}\right ) c_{2} +O\left (t^{6}\right ) \] Replacing \(t\) in the above with the original independent variable \(xs\)using \(t = x -2\) results in \[ \text {Expression too large to display} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} \text {Expression too large to display} \\ \end{align*}

Veriﬁcation of solutions

\[ \text {Expression too large to display} \] 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 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]

\[ y \left (x \right ) = \left (1+\frac {{\mathrm e}^{4} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}-\frac {2 \left ({\mathrm e}^{2}+4 \,{\mathrm e}^{4}\right ) {\mathrm e}^{2} \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {\left ({\mathrm e}^{2}+12 \,{\mathrm e}^{4}+\frac {33 \,{\mathrm e}^{6}}{2}+\frac {{\mathrm e}^{10}}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {2 \,{\mathrm e}^{2} \left ({\mathrm e}^{2}+\frac {53 \,{\mathrm e}^{4}}{2}+98 \,{\mathrm e}^{6}+79 \,{\mathrm e}^{8}+3 \,{\mathrm e}^{10}+\frac {5 \,{\mathrm e}^{12}}{2}\right ) \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) y \left (2\right )+\left (x -2-\frac {4 \,{\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}-\frac {2 \left (-\frac {31 \,{\mathrm e}^{2}}{2}-\frac {{\mathrm e}^{6}}{2}-4\right ) {\mathrm e}^{2} \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {\left (-47 \,{\mathrm e}^{2}-65 \,{\mathrm e}^{4}-{\mathrm e}^{6}-\frac {7 \,{\mathrm e}^{8}}{2}-\frac {7}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {2 \,{\mathrm e}^{2} \left (-\frac {205 \,{\mathrm e}^{2}}{2}-\frac {1537 \,{\mathrm e}^{4}}{4}-\frac {1249 \,{\mathrm e}^{6}}{4}-\frac {85 \,{\mathrm e}^{8}}{4}-17 \,{\mathrm e}^{10}+\frac {{\mathrm e}^{12}}{4}-\frac {{\mathrm e}^{14}}{4}-\frac {11}{4}\right ) \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

\[ y(x)\to c_2 \left (-\frac {1}{60} \left (-6 \text {csch}(2)+7 \coth (2) \text {csch}(2)+12 \coth ^2(2) \text {csch}(2)-12 \coth ^3(2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{20} \left (e^2 \coth (2) \text {csch}(2)-e^2 \coth ^2(2) \text {csch}(2)\right ) (x-2)^5+\frac {4}{15} \text {csch}(2) \left (-\text {csch}(2)-4 \coth (2) \text {csch}(2)+4 \coth ^2(2) \text {csch}(2)\right ) (x-2)^5+\frac {1}{6} \text {csch}(2) \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{40} (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (x-2)^5+\frac {2}{5} \text {csch}^2(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (x-2)^5+\frac {1}{40} e^2 \text {csch}(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (x-2)^5-\frac {2}{5} \text {csch}^2(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^5-\frac {1}{120} e^2 \text {csch}(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^5+\frac {32}{15} \text {csch}^4(2) (x-2)^5+\frac {2}{5} e^2 \text {csch}^3(2) (x-2)^5+\frac {1}{120} e^4 \text {csch}^2(2) (x-2)^5-\frac {1}{12} \left (-\text {csch}(2)-4 \coth (2) \text {csch}(2)+4 \coth ^2(2) \text {csch}(2)\right ) (x-2)^4-\frac {1}{12} \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^4+\frac {1}{2} \text {csch}(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^4-\frac {8}{3} \text {csch}^3(2) (x-2)^4-\frac {1}{3} e^2 \text {csch}^2(2) (x-2)^4-\frac {1}{6} (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^3+\frac {8}{3} \text {csch}^2(2) (x-2)^3+\frac {1}{6} e^2 \text {csch}(2) (x-2)^3-2 \text {csch}(2) (x-2)^2+x-2\right )+c_1 \left (-\frac {1}{60} \left (e^2 \text {csch}(2)-e^2 \coth (2) \text {csch}(2)-3 e^2 \coth ^2(2) \text {csch}(2)+3 e^2 \coth ^3(2) \text {csch}(2)\right ) (x-2)^5+\frac {1}{15} \text {csch}(2) \left (e^2 \coth (2) \text {csch}(2)-e^2 \coth ^2(2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{20} e^2 \text {csch}(2) \left (-\text {csch}(2)-4 \coth (2) \text {csch}(2)+4 \coth ^2(2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{40} (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^5-\frac {2}{15} \text {csch}^2(2) \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{30} e^2 \text {csch}(2) \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^5-\frac {1}{10} e^2 \text {csch}^2(2) (-4 \text {csch}(2)+4 \coth (2) \text {csch}(2)) (x-2)^5+\frac {1}{15} e^2 \text {csch}^2(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^5-\frac {8}{15} e^2 \text {csch}^4(2) (x-2)^5-\frac {1}{15} e^4 \text {csch}^3(2) (x-2)^5-\frac {1}{12} \left (e^2 \coth (2) \text {csch}(2)-e^2 \coth ^2(2) \text {csch}(2)\right ) (x-2)^4+\frac {1}{6} \text {csch}(2) \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^4-\frac {1}{12} e^2 \text {csch}(2) (4 \text {csch}(2)-4 \coth (2) \text {csch}(2)) (x-2)^4+\frac {2}{3} e^2 \text {csch}^3(2) (x-2)^4+\frac {1}{24} e^4 \text {csch}^2(2) (x-2)^4-\frac {1}{6} \left (-e^2 \text {csch}(2)+e^2 \coth (2) \text {csch}(2)\right ) (x-2)^3-\frac {2}{3} e^2 \text {csch}^2(2) (x-2)^3+\frac {1}{2} e^2 \text {csch}(2) (x-2)^2+1\right ) \]

24.4 problem 34.5 (d)