24.5 problem 34.5 (e)

Internal problem ID [13948]
Internal file name [OUTPUT/13120_Friday_February_23_2024_06_55_15_AM_40637694/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 (e).
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"

Maple gives the following as the ode type

[[_2nd_order, _with_linear_symmetries]]

\[ \boxed {\sinh \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \sin \left (x \right )=0} \] 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 \[ \sinh \left (2+t \right ) \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+\left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right )-\sin \left (2+t \right ) y \left (t \right ) = 0 \] 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 )+\sin \left (2+t \right ) y \left (t \right )-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 \\ &= \operatorname {csch}\left (2+t \right ) \left (\left (-\sin \left (2+t \right ) y \left (t \right )+\left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right )\right ) \left (2+t \right )^{2} \operatorname {csch}\left (2+t \right )+\left (-\coth \left (2+t \right ) y \left (t \right )+\frac {d}{d t}y \left (t \right )\right ) \sin \left (2+t \right )+\left (\frac {d}{d t}y \left (t \right )\right ) \left (2+t \right )^{2} \coth \left (2+t \right )+\cos \left (2+t \right ) y \left (t \right )-2 \left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )\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 \\ &= -\left (\left (-\sin \left (2+t \right ) y \left (t \right )+\left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right )\right ) \left (2+t \right )^{4} \operatorname {csch}\left (2+t \right )^{2}+\left (-\sin \left (2+t \right )^{2} y \left (t \right )-3 \left (2+t \right ) \left (y \left (t \right ) \left (2+t \right ) \coth \left (2+t \right )+\frac {2 \left (-t -2\right ) \left (\frac {d}{d t}y \left (t \right )\right )}{3}-\frac {4 y \left (t \right )}{3}\right ) \sin \left (2+t \right )+3 \left (2+t \right )^{2} \left (\left (\frac {d}{d t}y \left (t \right )\right ) \left (2+t \right )^{2} \coth \left (2+t \right )+\frac {\cos \left (2+t \right ) y \left (t \right )}{3}-2 \left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )\right )\right ) \operatorname {csch}\left (2+t \right )+2 \left (-y \left (t \right ) \coth \left (2+t \right )^{2}+\left (\frac {d}{d t}y \left (t \right )\right ) \coth \left (2+t \right )+y \left (t \right )\right ) \sin \left (2+t \right )+2 \left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right ) \coth \left (2+t \right )^{2}+2 \left (\cos \left (2+t \right ) y \left (t \right )-2 \left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )\right ) \coth \left (2+t \right )-\left (\frac {d}{d t}y \left (t \right )\right ) \left (t^{2}+4 t +2 \cos \left (2+t \right )+2\right )\right ) \operatorname {csch}\left (2+t \right )\\ 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 \\ &= \operatorname {csch}\left (2+t \right ) \left (\left (-\sin \left (2+t \right ) y \left (t \right )+\left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right )\right ) \left (2+t \right )^{6} \operatorname {csch}\left (2+t \right )^{3}+6 \left (2+t \right )^{2} \left (\left (-\sin \left (2+t \right ) y \left (t \right )+\left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right )\right ) \left (2+t \right )^{2} \coth \left (2+t \right )-\frac {\sin \left (2+t \right )^{2} y \left (t \right )}{3}+\frac {\left (2+t \right ) \left (\left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )+\frac {10 y \left (t \right )}{3}\right ) \sin \left (2+t \right )}{2}-2 \left (2+t \right )^{2} \left (-\frac {\cos \left (2+t \right ) y \left (t \right )}{12}+\left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )\right )\right ) \operatorname {csch}\left (2+t \right )^{2}+\left (11 \left (-\sin \left (2+t \right ) y \left (t \right )+\left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right )\right ) \left (2+t \right )^{2} \coth \left (2+t \right )^{2}+\left (-4 \sin \left (2+t \right )^{2} y \left (t \right )+9 \left (2+t \right ) \left (\left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )+2 y \left (t \right )\right ) \sin \left (2+t \right )-28 \left (2+t \right )^{2} \left (-\frac {5 \cos \left (2+t \right ) y \left (t \right )}{28}+\left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )\right )\right ) \coth \left (2+t \right )+\left (\frac {d}{d t}y \left (t \right )\right ) \sin \left (2+t \right )^{2}+\left (4 \cos \left (2+t \right ) y \left (t \right )+\left (-8 t -16\right ) \left (\frac {d}{d t}y \left (t \right )\right )+5 y \left (t \right ) \left (t^{2}+4 t +\frac {14}{5}\right )\right ) \sin \left (2+t \right )-4 \left (2+t \right ) \left (\left (\left (\frac {5 t}{4}+\frac {5}{2}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+\frac {3 y \left (t \right )}{2}\right ) \cos \left (2+t \right )+\left (\frac {d}{d t}y \left (t \right )\right ) \left (2+t \right ) \left (t^{2}+4 t -1\right )\right )\right ) \operatorname {csch}\left (2+t \right )+\left (-6 \sin \left (2+t \right ) y \left (t \right )+6 \left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right )\right ) \coth \left (2+t \right )^{3}+\left (6 \sin \left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )+6 \cos \left (2+t \right ) y \left (t \right )-12 \left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )\right ) \coth \left (2+t \right )^{2}+\left (8 \sin \left (2+t \right ) y \left (t \right )-5 \left (\frac {d}{d t}y \left (t \right )\right ) \left (t^{2}+4 t +\frac {6 \cos \left (2+t \right )}{5}+\frac {14}{5}\right )\right ) \coth \left (2+t \right )-6 \sin \left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )-4 \cos \left (2+t \right ) y \left (t \right )+6 \left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )\right )\\ 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 (\sin \left (2\right ) y \left (0\right )-4 y^{\prime }\left (0\right )\right ) {\mathrm e}^{2}}{{\mathrm e}^{4}-1}\\ F_1 &= \frac {4 \,{\mathrm e}^{2} \sin \left (2\right ) y^{\prime }\left (0\right )+16 \sin \left (2\right ) {\mathrm e}^{4} y \left (0\right )-2 \sin \left (2\right ) {\mathrm e}^{-2} y \left (0\right )-2 \sin \left (2\right ) {\mathrm e}^{-2} y^{\prime }\left (0\right )+2 \sin \left (2\right ) {\mathrm e}^{6} y \left (0\right )-2 \sin \left (2\right ) {\mathrm e}^{6} y^{\prime }\left (0\right )+4 y \left (0\right ) {\mathrm e}^{2} \cos \left (2\right )-2 \,{\mathrm e}^{-2} \cos \left (2\right ) y \left (0\right )-2 \cos \left (2\right ) {\mathrm e}^{6} y \left (0\right )-16 \sin \left (2\right ) y \left (0\right )-16 \,{\mathrm e}^{2} y^{\prime }\left (0\right )-64 \,{\mathrm e}^{4} y^{\prime }\left (0\right )+16 y^{\prime }\left (0\right ) {\mathrm e}^{-2}+64 y^{\prime }\left (0\right )}{-{\mathrm e}^{8}+3 \,{\mathrm e}^{4}-3+{\mathrm e}^{-4}}\\ F_2 &= \frac {2 y \left (0\right ) {\mathrm e}^{-2}+28 y^{\prime }\left (0\right ) {\mathrm e}^{-4}+6 y \left (0\right ) {\mathrm e}^{2} \cos \left (4\right )-6 \,{\mathrm e}^{6} y \left (0\right ) \cos \left (4\right )+496 y^{\prime }\left (0\right )+384 y^{\prime }\left (0\right ) {\mathrm e}^{-2}+6 y \left (0\right ) {\mathrm e}^{6}-80 \sin \left (2\right ) {\mathrm e}^{-2} y \left (0\right )-48 \sin \left (2\right ) {\mathrm e}^{6} y \left (0\right )-16 \,{\mathrm e}^{-2} \cos \left (2\right ) y \left (0\right )-48 \cos \left (2\right ) {\mathrm e}^{6} y \left (0\right )-768 \,{\mathrm e}^{2} y^{\prime }\left (0\right )+8 y^{\prime }\left (0\right ) \sin \left (2\right )+16 \cos \left (2\right ) y^{\prime }\left (0\right )-32 \sin \left (2\right ) {\mathrm e}^{-2} y^{\prime }\left (0\right )-96 \sin \left (2\right ) {\mathrm e}^{6} y^{\prime }\left (0\right )-6 y \left (0\right ) {\mathrm e}^{2}-144 \sin \left (2\right ) {\mathrm e}^{8} y \left (0\right )-16 \sin \left (2\right ) {\mathrm e}^{10} y \left (0\right )-4 \,{\mathrm e}^{-4} \cos \left (2\right ) y \left (0\right )-8 \cos \left (2\right ) {\mathrm e}^{8} y \left (0\right )+16 \cos \left (2\right ) {\mathrm e}^{10} y \left (0\right )+4 \cos \left (2\right ) {\mathrm e}^{12} y \left (0\right )+32 \sin \left (2\right ) {\mathrm e}^{10} y^{\prime }\left (0\right )+4 \sin \left (2\right ) {\mathrm e}^{12} y^{\prime }\left (0\right )-4 \,{\mathrm e}^{-4} \cos \left (2\right ) y^{\prime }\left (0\right )+16 \cos \left (2\right ) {\mathrm e}^{8} y^{\prime }\left (0\right )-4 \cos \left (2\right ) {\mathrm e}^{12} y^{\prime }\left (0\right )+8 \cos \left (2\right ) y \left (0\right )-1080 \,{\mathrm e}^{4} y^{\prime }\left (0\right )-144 \sin \left (2\right ) y \left (0\right )+288 \sin \left (2\right ) {\mathrm e}^{4} y \left (0\right )+560 y^{\prime }\left (0\right ) {\mathrm e}^{8}+384 y^{\prime }\left (0\right ) {\mathrm e}^{6}-2 y \left (0\right ) {\mathrm e}^{10}-4 \sin \left (2\right ) {\mathrm e}^{-4} y^{\prime }\left (0\right )-8 \sin \left (2\right ) {\mathrm e}^{8} y^{\prime }\left (0\right )-2 \,{\mathrm e}^{-2} y \left (0\right ) \cos \left (4\right )+2 \,{\mathrm e}^{10} y \left (0\right ) \cos \left (4\right )+96 \,{\mathrm e}^{2} \sin \left (2\right ) y^{\prime }\left (0\right )-4 y^{\prime }\left (0\right ) {\mathrm e}^{12}+144 y \left (0\right ) {\mathrm e}^{2} \sin \left (2\right )+48 y \left (0\right ) {\mathrm e}^{2} \cos \left (2\right )-24 \,{\mathrm e}^{4} y^{\prime }\left (0\right ) \cos \left (2\right )}{-{\mathrm e}^{14}+5 \,{\mathrm e}^{10}-10 \,{\mathrm e}^{6}+10 \,{\mathrm e}^{2}-5 \,{\mathrm e}^{-2}+{\mathrm e}^{-6}}\\ F_3 &= \text {Expression too large to display}\\ F_4 &= \text {Expression too large to display} \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 )+\sin \left (2+t \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right )-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 \(-\sin \left (2+t \right )\) as Taylor series around \(t=0\) and keeping only the first \(6\) terms gives \begin {align*} -\sin \left (2+t \right ) &= -\sin \left (2\right )-\cos \left (2\right ) t +\frac {\sin \left (2\right ) t^{2}}{2}+\frac {\cos \left (2\right ) t^{3}}{6}-\frac {\sin \left (2\right ) t^{4}}{24}-\frac {t^{5} \cos \left (2\right )}{120}+\frac {\sin \left (2\right ) t^{6}}{720} + \dots \\ &= -\sin \left (2\right )-\cos \left (2\right ) t +\frac {\sin \left (2\right ) t^{2}}{2}+\frac {\cos \left (2\right ) t^{3}}{6}-\frac {\sin \left (2\right ) t^{4}}{24}-\frac {t^{5} \cos \left (2\right )}{120}+\frac {\sin \left (2\right ) t^{6}}{720} \end {align*}

Hence the ODE in Eq (1) becomes \[ \left (\sinh \left (2\right )+\cosh \left (2\right ) t +\frac {\sinh \left (2\right ) t^{2}}{2}+\frac {\cosh \left (2\right ) t^{3}}{6}+\frac {\sinh \left (2\right ) t^{4}}{24}+\frac {\cosh \left (2\right ) t^{5}}{120}+\frac {\sinh \left (2\right ) t^{6}}{720}\right ) \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\left (t^{2}+4 t +4\right ) \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\right )+\left (-\sin \left (2\right )-\cos \left (2\right ) t +\frac {\sin \left (2\right ) t^{2}}{2}+\frac {\cos \left (2\right ) t^{3}}{6}-\frac {\sin \left (2\right ) t^{4}}{24}-\frac {t^{5} \cos \left (2\right )}{120}+\frac {\sin \left (2\right ) t^{6}}{720}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right ) = 0 \] Expanding the first term in (1) gives \[ \sinh \left (2\right )\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\cosh \left (2\right ) t \cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\sinh \left (2\right ) t^{2}}{2}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\cosh \left (2\right ) t^{3}}{6}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\sinh \left (2\right ) t^{4}}{24}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\cosh \left (2\right ) t^{5}}{120}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\sinh \left (2\right ) t^{6}}{720}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\left (t^{2}+4 t +4\right ) \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\right )+\left (-\sin \left (2\right )-\cos \left (2\right ) t +\frac {\sin \left (2\right ) t^{2}}{2}+\frac {\cos \left (2\right ) t^{3}}{6}-\frac {\sin \left (2\right ) t^{4}}{24}-\frac {t^{5} \cos \left (2\right )}{120}+\frac {\sin \left (2\right ) t^{6}}{720}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right ) = 0 \] 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} \sin \left (2\right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-t^{1+n} a_{n} \cos \left (2\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +2} a_{n} \sin \left (2\right )}{2}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +3} a_{n} \cos \left (2\right )}{6}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +4} a_{n} \sin \left (2\right )}{24}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +5} a_{n} \cos \left (2\right )}{120}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +6} a_{n} \sin \left (2\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 }}\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} \cos \left (2\right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \cos \left (2\right ) t^{n}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +2} a_{n} \sin \left (2\right )}{2} &= \moverset {\infty }{\munderset {n =2}{\sum }}\frac {a_{n -2} \sin \left (2\right ) t^{n}}{2} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +3} a_{n} \cos \left (2\right )}{6} &= \moverset {\infty }{\munderset {n =3}{\sum }}\frac {a_{n -3} \cos \left (2\right ) t^{n}}{6} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +4} a_{n} \sin \left (2\right )}{24}\right ) &= \moverset {\infty }{\munderset {n =4}{\sum }}\left (-\frac {a_{n -4} \sin \left (2\right ) t^{n}}{24}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\frac {t^{n +5} a_{n} \cos \left (2\right )}{120}\right ) &= \moverset {\infty }{\munderset {n =5}{\sum }}\left (-\frac {a_{n -5} \cos \left (2\right ) t^{n}}{120}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\frac {t^{n +6} a_{n} \sin \left (2\right )}{720} &= \moverset {\infty }{\munderset {n =6}{\sum }}\frac {a_{n -6} \sin \left (2\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} \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} \sin \left (2\right )\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \cos \left (2\right ) t^{n}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {a_{n -2} \sin \left (2\right ) t^{n}}{2}\right )+\left (\moverset {\infty }{\munderset {n =3}{\sum }}\frac {a_{n -3} \cos \left (2\right ) t^{n}}{6}\right )+\moverset {\infty }{\munderset {n =4}{\sum }}\left (-\frac {a_{n -4} \sin \left (2\right ) t^{n}}{24}\right )+\moverset {\infty }{\munderset {n =5}{\sum }}\left (-\frac {a_{n -5} \cos \left (2\right ) t^{n}}{120}\right )+\left (\moverset {\infty }{\munderset {n =6}{\sum }}\frac {a_{n -6} \sin \left (2\right ) t^{n}}{720}\right ) = 0 \end{equation} \(n=0\) gives \[ 2 a_{2} \sinh \left (2\right )+4 a_{1}-a_{0} \sin \left (2\right )=0 \] \[ a_{2} = \frac {a_{0} \sin \left (2\right )-4 a_{1}}{2 \sinh \left (2\right )} \] \(n=1\) gives \[ 2 a_{2} \cosh \left (2\right )+6 a_{3} \sinh \left (2\right )+4 a_{1}+8 a_{2}-a_{1} \sin \left (2\right )-a_{0} \cos \left (2\right )=0 \] Which after substituting earlier equations, simplifies to \[ a_{3} = \frac {a_{1} \sin \left (2\right ) \sinh \left (2\right )-\cosh \left (2\right ) a_{0} \sin \left (2\right )+a_{0} \cos \left (2\right ) \sinh \left (2\right )-4 a_{0} \sin \left (2\right )-4 a_{1} \sinh \left (2\right )+4 \cosh \left (2\right ) a_{1}+16 a_{1}}{6 \sinh \left (2\right )^{2}} \] \(n=2\) gives \[ a_{2} \sinh \left (2\right )+6 a_{3} \cosh \left (2\right )+12 a_{4} \sinh \left (2\right )+a_{1}+8 a_{2}+12 a_{3}-a_{2} \sin \left (2\right )-a_{1} \cos \left (2\right )+\frac {a_{0} \sin \left (2\right )}{2}=0 \] Which after substituting earlier equations, simplifies to \[ a_{4} = \frac {\sin \left (2\right )^{2} a_{0} \sinh \left (2\right )-2 a_{0} \sin \left (2\right ) \sinh \left (2\right )^{2}-2 \cosh \left (2\right ) a_{1} \sin \left (2\right ) \sinh \left (2\right )+2 \cosh \left (2\right )^{2} a_{0} \sin \left (2\right )+2 a_{1} \cos \left (2\right ) \sinh \left (2\right )^{2}-2 \cosh \left (2\right ) a_{0} \cos \left (2\right ) \sinh \left (2\right )-8 a_{0} \sin \left (2\right ) \sinh \left (2\right )-8 a_{1} \sin \left (2\right ) \sinh \left (2\right )+12 \cosh \left (2\right ) a_{0} \sin \left (2\right )-4 a_{0} \cos \left (2\right ) \sinh \left (2\right )+2 a_{1} \sinh \left (2\right )^{2}+8 \cosh \left (2\right ) a_{1} \sinh \left (2\right )-8 \cosh \left (2\right )^{2} a_{1}+16 a_{0} \sin \left (2\right )+48 a_{1} \sinh \left (2\right )-48 \cosh \left (2\right ) a_{1}-64 a_{1}}{24 \sinh \left (2\right )^{3}} \] \(n=3\) gives \[ \frac {a_{2} \cosh \left (2\right )}{3}+3 a_{3} \sinh \left (2\right )+12 a_{4} \cosh \left (2\right )+20 a_{5} \sinh \left (2\right )+2 a_{2}+12 a_{3}+16 a_{4}-a_{3} \sin \left (2\right )-a_{2} \cos \left (2\right )+\frac {a_{1} \sin \left (2\right )}{2}+\frac {a_{0} \cos \left (2\right )}{6}=0 \] Which after substituting earlier equations, simplifies to \[ a_{5} = \frac {-384 a_{1} \sinh \left (2\right )+384 \cosh \left (2\right ) a_{1}-8 \sin \left (2\right )^{2} a_{0} \sinh \left (2\right )+14 a_{0} \sin \left (2\right ) \sinh \left (2\right )^{2}-44 \cosh \left (2\right )^{2} a_{0} \sin \left (2\right )-20 a_{1} \cos \left (2\right ) \sinh \left (2\right )^{2}+80 a_{0} \sin \left (2\right ) \sinh \left (2\right )-224 \cosh \left (2\right ) a_{1} \sinh \left (2\right )+12 a_{1} \sinh \left (2\right )^{3}+24 \cosh \left (2\right )^{3} a_{1}+256 a_{1}-4 \cosh \left (2\right ) \sin \left (2\right )^{2} a_{0} \sinh \left (2\right )+4 \sin \left (2\right ) a_{0} \cos \left (2\right ) \sinh \left (2\right )^{2}+8 \cosh \left (2\right ) a_{0} \sin \left (2\right ) \sinh \left (2\right )^{2}+6 \cosh \left (2\right )^{2} a_{1} \sin \left (2\right ) \sinh \left (2\right )-6 \cosh \left (2\right ) a_{1} \cos \left (2\right ) \sinh \left (2\right )^{2}+6 \cosh \left (2\right )^{2} a_{0} \cos \left (2\right ) \sinh \left (2\right )+36 \cosh \left (2\right ) a_{0} \sin \left (2\right ) \sinh \left (2\right )+48 a_{1} \sin \left (2\right ) \sinh \left (2\right )-96 \cosh \left (2\right ) a_{0} \sin \left (2\right )+16 a_{0} \cos \left (2\right ) \sinh \left (2\right )+16 a_{1} \sinh \left (2\right )^{2}+176 \cosh \left (2\right )^{2} a_{1}+\sin \left (2\right )^{2} a_{1} \sinh \left (2\right )^{2}-6 a_{1} \sin \left (2\right ) \sinh \left (2\right )^{3}-6 \cosh \left (2\right )^{3} a_{0} \sin \left (2\right )-4 a_{0} \cos \left (2\right ) \sinh \left (2\right )^{3}-16 a_{1} \sin \left (2\right ) \sinh \left (2\right )^{2}-12 a_{0} \cos \left (2\right ) \sinh \left (2\right )^{2}-14 \cosh \left (2\right ) a_{1} \sinh \left (2\right )^{2}-24 \cosh \left (2\right )^{2} a_{1} \sinh \left (2\right )-64 a_{0} \sin \left (2\right )+36 \cosh \left (2\right ) a_{1} \sin \left (2\right ) \sinh \left (2\right )+20 \cosh \left (2\right ) a_{0} \cos \left (2\right ) \sinh \left (2\right )}{120 \sinh \left (2\right )^{4}} \] \(n=4\) gives \[ \frac {a_{2} \sinh \left (2\right )}{12}+a_{3} \cosh \left (2\right )+6 a_{4} \sinh \left (2\right )+20 a_{5} \cosh \left (2\right )+30 a_{6} \sinh \left (2\right )+3 a_{3}+16 a_{4}+20 a_{5}-a_{4} \sin \left (2\right )-a_{3} \cos \left (2\right )+\frac {a_{2} \sin \left (2\right )}{2}+\frac {a_{1} \cos \left (2\right )}{6}-\frac {a_{0} \sin \left (2\right )}{24}=0 \] Which after substituting earlier equations, simplifies to \[ \text {Expression too large to display} \] \(n=5\) gives \[ \frac {a_{2} \cosh \left (2\right )}{60}+\frac {a_{3} \sinh \left (2\right )}{4}+2 a_{4} \cosh \left (2\right )+10 a_{5} \sinh \left (2\right )+30 a_{6} \cosh \left (2\right )+42 a_{7} \sinh \left (2\right )+4 a_{4}+20 a_{5}+24 a_{6}-a_{5} \sin \left (2\right )-a_{4} \cos \left (2\right )+\frac {a_{3} \sin \left (2\right )}{2}+\frac {a_{2} \cos \left (2\right )}{6}-\frac {a_{1} \sin \left (2\right )}{24}-\frac {a_{0} \cos \left (2\right )}{120}=0 \] 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} \sin \left (2\right )-a_{n -1} \cos \left (2\right )+\frac {a_{n -2} \sin \left (2\right )}{2}+\frac {a_{n -3} \cos \left (2\right )}{6}-\frac {a_{n -4} \sin \left (2\right )}{24}-\frac {a_{n -5} \cos \left (2\right )}{120}+\frac {a_{n -6} \sin \left (2\right )}{720} = 0 \end{equation} Solving for \(a_{n +2}\), gives \begin{align*} \tag{5} a_{n +2}&= \frac {-2880 n a_{n}+720 a_{n} \sin \left (2\right )+30 a_{n -4} \sin \left (2\right )-2880 a_{1+n}-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 a_{n -2} \sin \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}-a_{n -6} \sin \left (2\right )+6 a_{n -5} \cos \left (2\right )+720 a_{n -1}+720 a_{n -1} \cos \left (2\right )-120 a_{n -3} \cos \left (2\right )}{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 \sin \left (2\right )-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 {\sin \left (2\right ) a_{n -6}}{720 \sinh \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\cos \left (2\right ) 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 \sin \left (2\right )-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 \cos \left (2\right )-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 \sin \left (2\right )-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 \cos \left (2\right )-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 \[ \text {Expression too large to display}+\dots \] Collecting terms, the solution becomes \begin{equation} \tag{3} y \left (t \right ) = \left (1+\frac {\sin \left (2\right ) t^{2}}{2 \sinh \left (2\right )}+\frac {\left (-\sin \left (2\right ) \cosh \left (2\right )+\cos \left (2\right ) \sinh \left (2\right )-4 \sin \left (2\right )\right ) t^{3}}{6 \sinh \left (2\right )^{2}}+\frac {\left (\sin \left (2\right )^{2} \sinh \left (2\right )-2 \sin \left (2\right ) \sinh \left (2\right )^{2}+2 \sin \left (2\right ) \cosh \left (2\right )^{2}-2 \cos \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )-8 \sin \left (2\right ) \sinh \left (2\right )+12 \sin \left (2\right ) \cosh \left (2\right )-4 \cos \left (2\right ) \sinh \left (2\right )+16 \sin \left (2\right )\right ) t^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left (-4 \sin \left (2\right )^{2} \sinh \left (2\right ) \cosh \left (2\right )+4 \sin \left (2\right ) \cos \left (2\right ) \sinh \left (2\right )^{2}+8 \sin \left (2\right ) \sinh \left (2\right )^{2} \cosh \left (2\right )-6 \sin \left (2\right ) \cosh \left (2\right )^{3}-4 \cos \left (2\right ) \sinh \left (2\right )^{3}+6 \cos \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )^{2}-8 \sin \left (2\right )^{2} \sinh \left (2\right )+14 \sin \left (2\right ) \sinh \left (2\right )^{2}+36 \sin \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )-44 \sin \left (2\right ) \cosh \left (2\right )^{2}-12 \cos \left (2\right ) \sinh \left (2\right )^{2}+20 \cos \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )+80 \sin \left (2\right ) \sinh \left (2\right )-96 \sin \left (2\right ) \cosh \left (2\right )+16 \cos \left (2\right ) \sinh \left (2\right )-64 \sin \left (2\right )\right ) t^{5}}{120 \sinh \left (2\right )^{4}}\right ) a_{0}+\left (t -\frac {2 t^{2}}{\sinh \left (2\right )}+\frac {\left (\sin \left (2\right ) \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 \sin \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )+2 \cos \left (2\right ) \sinh \left (2\right )^{2}-8 \sin \left (2\right ) \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 (\sin \left (2\right )^{2} \sinh \left (2\right )^{2}-6 \sin \left (2\right ) \sinh \left (2\right )^{3}+6 \sin \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )^{2}-6 \cos \left (2\right ) \sinh \left (2\right )^{2} \cosh \left (2\right )-16 \sin \left (2\right ) \sinh \left (2\right )^{2}+36 \sin \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )-20 \cos \left (2\right ) \sinh \left (2\right )^{2}+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 \sin \left (2\right ) \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 {\sin \left (2\right ) t^{2}}{2 \sinh \left (2\right )}+\frac {\left (-\sin \left (2\right ) \cosh \left (2\right )+\cos \left (2\right ) \sinh \left (2\right )-4 \sin \left (2\right )\right ) t^{3}}{6 \sinh \left (2\right )^{2}}+\frac {\left (\sin \left (2\right )^{2} \sinh \left (2\right )-2 \sin \left (2\right ) \sinh \left (2\right )^{2}+2 \sin \left (2\right ) \cosh \left (2\right )^{2}-2 \cos \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )-8 \sin \left (2\right ) \sinh \left (2\right )+12 \sin \left (2\right ) \cosh \left (2\right )-4 \cos \left (2\right ) \sinh \left (2\right )+16 \sin \left (2\right )\right ) t^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left (-4 \sin \left (2\right )^{2} \sinh \left (2\right ) \cosh \left (2\right )+4 \sin \left (2\right ) \cos \left (2\right ) \sinh \left (2\right )^{2}+8 \sin \left (2\right ) \sinh \left (2\right )^{2} \cosh \left (2\right )-6 \sin \left (2\right ) \cosh \left (2\right )^{3}-4 \cos \left (2\right ) \sinh \left (2\right )^{3}+6 \cos \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )^{2}-8 \sin \left (2\right )^{2} \sinh \left (2\right )+14 \sin \left (2\right ) \sinh \left (2\right )^{2}+36 \sin \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )-44 \sin \left (2\right ) \cosh \left (2\right )^{2}-12 \cos \left (2\right ) \sinh \left (2\right )^{2}+20 \cos \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )+80 \sin \left (2\right ) \sinh \left (2\right )-96 \sin \left (2\right ) \cosh \left (2\right )+16 \cos \left (2\right ) \sinh \left (2\right )-64 \sin \left (2\right )\right ) t^{5}}{120 \sinh \left (2\right )^{4}}\right ) c_{1} +\left (t -\frac {2 t^{2}}{\sinh \left (2\right )}+\frac {\left (\sin \left (2\right ) \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 \sin \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )+2 \cos \left (2\right ) \sinh \left (2\right )^{2}-8 \sin \left (2\right ) \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 (\sin \left (2\right )^{2} \sinh \left (2\right )^{2}-6 \sin \left (2\right ) \sinh \left (2\right )^{3}+6 \sin \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )^{2}-6 \cos \left (2\right ) \sinh \left (2\right )^{2} \cosh \left (2\right )-16 \sin \left (2\right ) \sinh \left (2\right )^{2}+36 \sin \left (2\right ) \sinh \left (2\right ) \cosh \left (2\right )-20 \cos \left (2\right ) \sinh \left (2\right )^{2}+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 \sin \left (2\right ) \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.

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 710

Order:=6; 
dsolve(sinh(x)*diff(y(x),x$2)+x^2*diff(y(x),x)-sin(x)*y(x)=0,y(x),type='series',x=2);
 

\[ y \left (x \right ) = \left (1+\frac {\sin \left (2\right ) {\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}+\frac {{\mathrm e}^{2} \left (\left (-8 \,{\mathrm e}^{2}-{\mathrm e}^{4}-1\right ) \sin \left (2\right )+\cos \left (2\right ) \left ({\mathrm e}^{4}-1\right )\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {2 \left (\frac {\left (-{\mathrm e}^{2}+{\mathrm e}^{6}\right ) \sin \left (2\right )^{2}}{4}+\left (5 \,{\mathrm e}^{2}+9 \,{\mathrm e}^{4}+{\mathrm e}^{6}\right ) \sin \left (2\right )+\cos \left (2\right ) \left ({\mathrm e}^{2}-{\mathrm e}^{6}-\frac {{\mathrm e}^{8}}{4}+\frac {1}{4}\right )\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}+\frac {2 \left (\left (\left ({\mathrm e}^{2}-2 \,{\mathrm e}^{6}+{\mathrm e}^{10}\right ) \sin \left (2\right )-8 \,{\mathrm e}^{2}-\frac {41 \,{\mathrm e}^{4}}{4}+6 \,{\mathrm e}^{6}+\frac {41 \,{\mathrm e}^{8}}{4}+2 \,{\mathrm e}^{10}+\frac {{\mathrm e}^{12}}{4}-\frac {1}{4}\right ) \cos \left (2\right )+\left (\left ({\mathrm e}^{2}+4 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{8}-{\mathrm e}^{10}\right ) \sin \left (2\right )-\frac {33 \,{\mathrm e}^{2}}{2}-\frac {365 \,{\mathrm e}^{4}}{4}-93 \,{\mathrm e}^{6}-\frac {45 \,{\mathrm e}^{8}}{4}+\frac {3 \,{\mathrm e}^{10}}{2}+\frac {{\mathrm e}^{12}}{4}+\frac {1}{4}\right ) \sin \left (2\right )\right ) {\mathrm e}^{2} \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 {{\mathrm e}^{2} \left (\left ({\mathrm e}^{4}-1\right ) \sin \left (2\right )+32 \,{\mathrm e}^{2}+8\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {2 \left (-\frac {7}{4}+\left (2 \,{\mathrm e}^{2}-2 \,{\mathrm e}^{6}-\frac {{\mathrm e}^{8}}{4}+\frac {1}{4}\right ) \sin \left (2\right )+\frac {\left (\frac {1}{2}-{\mathrm e}^{4}+\frac {{\mathrm e}^{8}}{2}\right ) \cos \left (2\right )}{2}-24 \,{\mathrm e}^{2}-\frac {69 \,{\mathrm e}^{4}}{2}+\frac {{\mathrm e}^{8}}{4}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}+\frac {2 \left (\left (-\frac {{\mathrm e}^{2}}{4}-\frac {{\mathrm e}^{10}}{4}+\frac {{\mathrm e}^{6}}{2}\right ) \cos \left (2\right )^{2}+\left (-\frac {3}{4}+\frac {3 \,{\mathrm e}^{4}}{4}+\frac {3 \,{\mathrm e}^{8}}{4}-\frac {3 \,{\mathrm e}^{12}}{4}-5 \,{\mathrm e}^{10}+10 \,{\mathrm e}^{6}-5 \,{\mathrm e}^{2}\right ) \cos \left (2\right )+\left (-13 \,{\mathrm e}^{2}-27 \,{\mathrm e}^{4}+8 \,{\mathrm e}^{6}+27 \,{\mathrm e}^{8}+5 \,{\mathrm e}^{10}\right ) \sin \left (2\right )+\frac {11}{4}+\frac {13 \,{\mathrm e}^{8}}{4}-\frac {{\mathrm e}^{12}}{4}-\frac {31 \,{\mathrm e}^{10}}{4}+\frac {1609 \,{\mathrm e}^{4}}{4}+\frac {417 \,{\mathrm e}^{2}}{4}+\frac {671 \,{\mathrm e}^{6}}{2}\right ) {\mathrm e}^{2} \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 ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 934

AsymptoticDSolveValue[Sinh[x]*y''[x]+x^2*y'[x]-Sin[x]*y[x]==0,y[x],{x,2,5}]
 

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