24.3 problem 34.5 (c)

Internal problem ID [13946]
Internal file name [OUTPUT/13118_Friday_February_23_2024_06_54_56_AM_54232141/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 (c).
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]]

\[ \boxed {\sin \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \,{\mathrm e}^{x}=0} \] 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 \[ \sin \left (2+t \right ) \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+\left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right )-y \left (t \right ) {\mathrm e}^{2+t} = 0 \] 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 )+4 t \left (\frac {d}{d t}y \left (t \right )\right )-y \left (t \right ) {\mathrm e}^{2+t}+4 \frac {d}{d t}y \left (t \right )}{\sin \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 (\left (-\cos \left (2+t \right ) y \left (t \right )+\left (\frac {d}{d t}y \left (t \right )+y \left (t \right )\right ) \sin \left (2+t \right )-\left (2+t \right )^{2} y \left (t \right )\right ) {\mathrm e}^{2+t}+\left (\frac {d}{d t}y \left (t \right )\right ) \left (2+t \right ) \left (\left (2+t \right ) \cos \left (2+t \right )+t^{3}+6 t^{2}+12 t -2 \sin \left (2+t \right )+8\right )\right ) \csc \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 \\ &= -\left (-y \left (t \right ) \sin \left (2+t \right ) {\mathrm e}^{2 t +4}+\left (-2 \left (\frac {d}{d t}y \left (t \right )\right ) \sin \left (2+t \right )^{2}+\left (\left (2 y \left (t \right )+2 \frac {d}{d t}y \left (t \right )\right ) \cos \left (2+t \right )+\left (2+t \right ) \left (\left (2 t +4\right ) \left (\frac {d}{d t}y \left (t \right )\right )+y \left (t \right ) \left (t +6\right )\right )\right ) \sin \left (2+t \right )-\left (3 \left (2+t \right )^{2} \cos \left (2+t \right )+t^{4}+8 t^{3}+24 t^{2}+32 t +18\right ) y \left (t \right )\right ) {\mathrm e}^{2+t}+\left (\frac {d}{d t}y \left (t \right )\right ) \left (-6 \left (2+t \right ) \left (\frac {2 \cos \left (2+t \right )}{3}+\left (2+t \right )^{2}\right ) \sin \left (2+t \right )+\left (t^{2}+4 t +2\right ) \cos \left (2+t \right )^{2}+3 \left (2+t \right )^{4} \cos \left (2+t \right )+\left (t^{2}+4 t +5\right ) \left (t^{4}+8 t^{3}+23 t^{2}+28 t +14\right )\right )\right ) \csc \left (2+t \right )^{3}\\ 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 \\ &= \left (-2 \sin \left (2+t \right ) \left (2 \cos \left (2+t \right ) y \left (t \right )+\left (-2 y \left (t \right )-\frac {\frac {d}{d t}y \left (t \right )}{2}\right ) \sin \left (2+t \right )+\left (2+t \right )^{2} y \left (t \right )\right ) {\mathrm e}^{2 t +4}+\left (2 y \left (t \right ) {\mathrm e}^{2+t}+\left (\frac {d}{d t}y \left (t \right )\right ) \left (t^{2}+4 t -2\right )\right ) \cos \left (2+t \right )^{3}+\left (\left (2 y \left (t \right ) {\mathrm e}^{2+t}-6 \left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right )\right ) \sin \left (2+t \right )-6 y \left (t \right ) \left (t^{2}+3 t +1\right ) {\mathrm e}^{2+t}+7 \left (\frac {d}{d t}y \left (t \right )\right ) \left (2+t \right )^{2} \left (t^{2}+4 t +\frac {8}{7}\right )\right ) \cos \left (2+t \right )^{2}+\left (-6 \left (\frac {d}{d t}y \left (t \right )\right ) \sin \left (2+t \right )^{2} {\mathrm e}^{2+t}-28 \left (2+t \right ) \left (\left (\left (-\frac {9 t}{28}-\frac {9}{14}\right ) \left (\frac {d}{d t}y \left (t \right )\right )+y \left (t \right ) \left (-\frac {5 t}{28}-1\right )\right ) {\mathrm e}^{2+t}+\left (2+t \right )^{2} \left (\frac {d}{d t}y \left (t \right )\right )\right ) \sin \left (2+t \right )-6 \left (t^{4}+8 t^{3}+24 t^{2}+32 t +\frac {52}{3}\right ) y \left (t \right ) {\mathrm e}^{2+t}+6 \left (t^{6}+12 t^{5}+60 t^{4}+160 t^{3}+\frac {1445}{6} t^{2}+\frac {586}{3} t +\frac {205}{3}\right ) \left (\frac {d}{d t}y \left (t \right )\right )\right ) \cos \left (2+t \right )-5 \left (2+t \right ) \left (\frac {d}{d t}y \left (t \right )\right ) {\mathrm e}^{2+t} \left (t +\frac {18}{5}\right ) \sin \left (2+t \right )^{2}+\left (\left (\left (3 t^{4}+24 t^{3}+72 t^{2}+96 t +54\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 +100\right )\right ) {\mathrm e}^{2+t}-12 \left (\frac {d}{d t}y \left (t \right )\right ) \left (2+t \right ) \left (t^{4}+8 t^{3}+24 t^{2}+32 t +\frac {33}{2}\right )\right ) \sin \left (2+t \right )-y \left (t \right ) \left (t^{6}+12 t^{5}+60 t^{4}+160 t^{3}+245 t^{2}+218 t +102\right ) {\mathrm e}^{2+t}+\left (\frac {d}{d t}y \left (t \right )\right ) \left (t^{6}+12 t^{5}+60 t^{4}+160 t^{3}+244 t^{2}+208 t +100\right ) \left (2+t \right )^{2}\right ) \csc \left (2+t \right )^{4}\\ 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 &= \csc \left (2\right ) \left (y \left (0\right ) {\mathrm e}^{2}-4 y^{\prime }\left (0\right )\right )\\ F_1 &= y \left (0\right ) {\mathrm e}^{2} \sin \left (2\right ) \csc \left (2\right )^{2}-y \left (0\right ) {\mathrm e}^{2} \cos \left (2\right ) \csc \left (2\right )^{2}+{\mathrm e}^{2} \sin \left (2\right ) \csc \left (2\right )^{2} y^{\prime }\left (0\right )-4 y \left (0\right ) {\mathrm e}^{2} \csc \left (2\right )^{2}-4 \sin \left (2\right ) \csc \left (2\right )^{2} y^{\prime }\left (0\right )+4 \cos \left (2\right ) \csc \left (2\right )^{2} y^{\prime }\left (0\right )+16 y^{\prime }\left (0\right ) \csc \left (2\right )^{2}\\ F_2 &= \frac {-y \left (0\right ) {\mathrm e}^{2} \sin \left (4\right )-{\mathrm e}^{2} \sin \left (4\right ) y^{\prime }\left (0\right )-12 y \left (0\right ) {\mathrm e}^{2} \sin \left (2\right )+12 y \left (0\right ) {\mathrm e}^{2} \cos \left (2\right )+\sin \left (2\right ) {\mathrm e}^{4} y \left (0\right )-8 \,{\mathrm e}^{2} \sin \left (2\right ) y^{\prime }\left (0\right )-{\mathrm e}^{2} \cos \left (4\right ) y^{\prime }\left (0\right )+4 \sin \left (4\right ) y^{\prime }\left (0\right )+18 y \left (0\right ) {\mathrm e}^{2}+{\mathrm e}^{2} y^{\prime }\left (0\right )+48 y^{\prime }\left (0\right ) \sin \left (2\right )-48 \cos \left (2\right ) y^{\prime }\left (0\right )-\cos \left (4\right ) y^{\prime }\left (0\right )-71 y^{\prime }\left (0\right )}{\sin \left (2\right )^{3}}\\ F_3 &= \frac {-12 \,{\mathrm e}^{2} \sin \left (2\right )^{2} \cos \left (2\right ) y^{\prime }\left (0\right )-72 \,{\mathrm e}^{2} \sin \left (2\right )^{2} y^{\prime }\left (0\right )+56 y \left (0\right ) {\mathrm e}^{2} \sin \left (4\right )-4 \,{\mathrm e}^{4} y \left (0\right ) \sin \left (4\right )+36 \,{\mathrm e}^{2} \sin \left (4\right ) y^{\prime }\left (0\right )+y \left (0\right ) {\mathrm e}^{2} \sin \left (6\right )+201 y \left (0\right ) {\mathrm e}^{2} \sin \left (2\right )-205 y \left (0\right ) {\mathrm e}^{2} \cos \left (2\right )+y \left (0\right ) {\mathrm e}^{2} \cos \left (6\right )-6 y \left (0\right ) {\mathrm e}^{2} \cos \left (4\right )-16 \sin \left (2\right ) {\mathrm e}^{4} y \left (0\right )-4 y \left (0\right ) {\mathrm e}^{4} \cos \left (4\right )+108 \,{\mathrm e}^{2} \sin \left (2\right ) y^{\prime }\left (0\right )-{\mathrm e}^{4} y^{\prime }\left (0\right ) \cos \left (4\right )-224 \sin \left (4\right ) y^{\prime }\left (0\right )-6 \sin \left (6\right ) y^{\prime }\left (0\right )-210 y \left (0\right ) {\mathrm e}^{2}+4 y \left (0\right ) {\mathrm e}^{4}-798 y^{\prime }\left (0\right ) \sin \left (2\right )+817 \cos \left (2\right ) y^{\prime }\left (0\right )-\cos \left (6\right ) y^{\prime }\left (0\right )+{\mathrm e}^{4} y^{\prime }\left (0\right )+32 \cos \left (4\right ) y^{\prime }\left (0\right )+832 y^{\prime }\left (0\right )}{2 \sin \left (2\right )^{4}}\\ F_4 &= \frac {60 \,{\mathrm e}^{4} y \left (0\right ) \sin \left (4\right )-42 y \left (0\right ) {\mathrm e}^{2} \cos \left (6\right )-74 y \left (0\right ) {\mathrm e}^{2} \sin \left (6\right )+84 y \left (0\right ) {\mathrm e}^{2} \cos \left (4\right )-y \left (0\right ) {\mathrm e}^{2} \cos \left (8\right )-{\mathrm e}^{6} y \left (0\right ) \cos \left (4\right )-6338 y^{\prime }\left (0\right )+320 \sin \left (6\right ) y^{\prime }\left (0\right )+140 \cos \left (6\right ) y^{\prime }\left (0\right )+y \left (0\right ) {\mathrm e}^{6}+6400 y^{\prime }\left (0\right ) \sin \left (2\right )-6860 \cos \left (2\right ) y^{\prime }\left (0\right )+1597 y \left (0\right ) {\mathrm e}^{2}+132 \sin \left (2\right ) {\mathrm e}^{4} y \left (0\right )+808 \,{\mathrm e}^{2} \sin \left (2\right )^{2} y^{\prime }\left (0\right )-120 \sin \left (2\right )^{2} {\mathrm e}^{4} y \left (0\right )+448 \,{\mathrm e}^{2} \sin \left (2\right )^{2} \cos \left (2\right ) y^{\prime }\left (0\right )-864 \,{\mathrm e}^{2} \sin \left (2\right ) y^{\prime }\left (0\right )-8 \sin \left (4\right ) {\mathrm e}^{2} \sin \left (2\right )^{2} y^{\prime }\left (0\right )+8 \,{\mathrm e}^{2} \sin \left (2\right )^{2} \cos \left (4\right ) y^{\prime }\left (0\right )-12 \sin \left (2\right )^{2} {\mathrm e}^{4} \cos \left (2\right ) y^{\prime }\left (0\right )-44 \sin \left (2\right )^{2} {\mathrm e}^{4} \cos \left (2\right ) y \left (0\right )+3288 \sin \left (4\right ) y^{\prime }\left (0\right )-384 \cos \left (4\right ) y^{\prime }\left (0\right )+28 \,{\mathrm e}^{2} \sin \left (2\right )^{3} y^{\prime }\left (0\right )+12 \sin \left (2\right )^{3} {\mathrm e}^{4} y^{\prime }\left (0\right )-24 \sin \left (2\right )^{2} {\mathrm e}^{4} y^{\prime }\left (0\right )-408 \,{\mathrm e}^{2} \sin \left (4\right ) y^{\prime }\left (0\right )-824 y \left (0\right ) {\mathrm e}^{2} \sin \left (4\right )-1618 y \left (0\right ) {\mathrm e}^{2} \sin \left (2\right )+1722 y \left (0\right ) {\mathrm e}^{2} \cos \left (2\right )+4 \sin \left (8\right ) y^{\prime }\left (0\right )+2 \cos \left (8\right ) y^{\prime }\left (0\right )}{2 \sin \left (2\right )^{5}} \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 )+4 t \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 \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\right )}{\sin \left (2+t \right )}\tag {1} \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*}

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 \[ \left (\sin \left (2\right )+\cos \left (2\right ) t -\frac {\sin \left (2\right ) t^{2}}{2}-\frac {\cos \left (2\right ) t^{3}}{6}+\frac {\sin \left (2\right ) t^{4}}{24}+\frac {t^{5} \cos \left (2\right )}{120}-\frac {\sin \left (2\right ) t^{6}}{720}\right ) \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\left (t^{2}+4 t +4\right ) \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\right )+\left (-{\mathrm e}^{2}-{\mathrm e}^{2} t -\frac {{\mathrm e}^{2} t^{2}}{2}-\frac {{\mathrm e}^{2} t^{3}}{6}-\frac {t^{4} {\mathrm e}^{2}}{24}-\frac {t^{5} {\mathrm e}^{2}}{120}-\frac {t^{6} {\mathrm e}^{2}}{720}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right ) = 0 \] Expanding the first term in (1) gives \[ \sin \left (2\right )\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\cos \left (2\right ) t \cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )-\frac {\sin \left (2\right ) t^{2}}{2}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )-\frac {\cos \left (2\right ) t^{3}}{6}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {\sin \left (2\right ) t^{4}}{24}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\frac {t^{5} \cos \left (2\right )}{120}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )-\frac {\sin \left (2\right ) t^{6}}{720}\cdot \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} t^{n -2}\right )+\left (t^{2}+4 t +4\right ) \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} t^{n -1}\right )+\left (-{\mathrm e}^{2}-{\mathrm e}^{2} t -\frac {{\mathrm e}^{2} t^{2}}{2}-\frac {{\mathrm e}^{2} t^{3}}{6}-\frac {t^{4} {\mathrm e}^{2}}{24}-\frac {t^{5} {\mathrm e}^{2}}{120}-\frac {t^{6} {\mathrm e}^{2}}{720}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right ) = 0 \] Expanding the third term in (1) gives \[ \text {Expression too large to display} \] Which simplifies to \begin{equation} \tag{2} \moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n \,t^{n +4} a_{n} \left (n -1\right ) \sin \left (2\right )}{720}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +3} a_{n} \left (n -1\right ) \cos \left (2\right )}{120}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +2} a_{n} \left (n -1\right ) \sin \left (2\right )}{24}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n \,t^{1+n} a_{n} \left (n -1\right ) \cos \left (2\right )}{6}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n a_{n} t^{n} \sin \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 ) \cos \left (2\right )\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}n \,t^{n -2} a_{n} \left (n -1\right ) \sin \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 }}\left (-\frac {n \,t^{n +4} a_{n} \left (n -1\right ) \sin \left (2\right )}{720}\right ) &= \moverset {\infty }{\munderset {n =6}{\sum }}\left (-\frac {\left (n -4\right ) a_{n -4} \left (n -5\right ) \sin \left (2\right ) t^{n}}{720}\right ) \\ \moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +3} a_{n} \left (n -1\right ) \cos \left (2\right )}{120} &= \moverset {\infty }{\munderset {n =5}{\sum }}\frac {\left (n -3\right ) a_{n -3} \left (n -4\right ) \cos \left (2\right ) t^{n}}{120} \\ \moverset {\infty }{\munderset {n =2}{\sum }}\frac {n \,t^{n +2} a_{n} \left (n -1\right ) \sin \left (2\right )}{24} &= \moverset {\infty }{\munderset {n =4}{\sum }}\frac {\left (n -2\right ) a_{n -2} \left (n -3\right ) \sin \left (2\right ) t^{n}}{24} \\ \moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n \,t^{1+n} a_{n} \left (n -1\right ) \cos \left (2\right )}{6}\right ) &= \moverset {\infty }{\munderset {n =3}{\sum }}\left (-\frac {\left (n -1\right ) a_{n -1} \left (n -2\right ) \cos \left (2\right ) t^{n}}{6}\right ) \\ \moverset {\infty }{\munderset {n =2}{\sum }}n \,t^{n -1} a_{n} \left (n -1\right ) \cos \left (2\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (1+n \right ) a_{1+n} n \cos \left (2\right ) t^{n} \\ \moverset {\infty }{\munderset {n =2}{\sum }}n \,t^{n -2} a_{n} \left (n -1\right ) \sin \left (2\right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +2\right ) a_{n +2} \left (1+n \right ) \sin \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} \moverset {\infty }{\munderset {n =6}{\sum }}\left (-\frac {\left (n -4\right ) a_{n -4} \left (n -5\right ) \sin \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 ) \cos \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 ) \sin \left (2\right ) t^{n}}{24}\right )+\moverset {\infty }{\munderset {n =3}{\sum }}\left (-\frac {\left (n -1\right ) a_{n -1} \left (n -2\right ) \cos \left (2\right ) t^{n}}{6}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {n a_{n} t^{n} \sin \left (2\right ) \left (n -1\right )}{2}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}\left (1+n \right ) a_{1+n} n \cos \left (2\right ) t^{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +2\right ) a_{n +2} \left (1+n \right ) \sin \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 \[ 2 a_{2} \sin \left (2\right )+4 a_{1}-a_{0} {\mathrm e}^{2}=0 \] \[ a_{2} = \frac {a_{0} {\mathrm e}^{2}-4 a_{1}}{2 \sin \left (2\right )} \] \(n=1\) gives \[ 2 a_{2} \cos \left (2\right )+6 a_{3} \sin \left (2\right )+4 a_{1}+8 a_{2}-a_{1} {\mathrm e}^{2}-a_{0} {\mathrm e}^{2}=0 \] Which after substituting earlier equations, simplifies to \[ a_{3} = \frac {a_{0} {\mathrm e}^{2} \sin \left (2\right )+a_{1} {\mathrm e}^{2} \sin \left (2\right )-\cos \left (2\right ) a_{0} {\mathrm e}^{2}-4 a_{1} \sin \left (2\right )+4 \cos \left (2\right ) a_{1}-4 a_{0} {\mathrm e}^{2}+16 a_{1}}{6 \sin \left (2\right )^{2}} \] \(n=2\) gives \[ -a_{2} \sin \left (2\right )+6 a_{3} \cos \left (2\right )+12 a_{4} \sin \left (2\right )+a_{1}+8 a_{2}+12 a_{3}-a_{2} {\mathrm e}^{2}-a_{1} {\mathrm e}^{2}-\frac {a_{0} {\mathrm e}^{2}}{2}=0 \] Which after substituting earlier equations, simplifies to \[ a_{4} = \frac {2 a_{0} {\mathrm e}^{2} \sin \left (2\right )^{2}+2 a_{1} {\mathrm e}^{2} \sin \left (2\right )^{2}-2 \cos \left (2\right ) a_{0} {\mathrm e}^{2} \sin \left (2\right )-2 \cos \left (2\right ) a_{1} {\mathrm e}^{2} \sin \left (2\right )+{\mathrm e}^{4} a_{0} \sin \left (2\right )+2 \cos \left (2\right )^{2} a_{0} {\mathrm e}^{2}-6 a_{1} \sin \left (2\right )^{2}+8 \cos \left (2\right ) a_{1} \sin \left (2\right )-12 a_{0} {\mathrm e}^{2} \sin \left (2\right )-8 a_{1} {\mathrm e}^{2} \sin \left (2\right )-8 \cos \left (2\right )^{2} a_{1}+12 \cos \left (2\right ) a_{0} {\mathrm e}^{2}+48 a_{1} \sin \left (2\right )-48 \cos \left (2\right ) a_{1}+16 a_{0} {\mathrm e}^{2}-64 a_{1}}{24 \sin \left (2\right )^{3}} \] \(n=3\) gives \[ -\frac {a_{2} \cos \left (2\right )}{3}-3 a_{3} \sin \left (2\right )+12 a_{4} \cos \left (2\right )+20 a_{5} \sin \left (2\right )+2 a_{2}+12 a_{3}+16 a_{4}-a_{3} {\mathrm e}^{2}-a_{2} {\mathrm e}^{2}-\frac {a_{1} {\mathrm e}^{2}}{2}-\frac {a_{0} {\mathrm e}^{2}}{6}=0 \] Which after substituting earlier equations, simplifies to \[ a_{5} = \frac {-64 a_{0} {\mathrm e}^{2}-8 \cos \left (2\right ) a_{0} {\mathrm e}^{2} \sin \left (2\right )^{2}-6 \cos \left (2\right ) a_{1} {\mathrm e}^{2} \sin \left (2\right )^{2}+6 \cos \left (2\right )^{2} a_{0} {\mathrm e}^{2} \sin \left (2\right )+6 \cos \left (2\right )^{2} a_{1} {\mathrm e}^{2} \sin \left (2\right )-4 \cos \left (2\right ) {\mathrm e}^{4} a_{0} \sin \left (2\right )+56 \cos \left (2\right ) a_{0} {\mathrm e}^{2} \sin \left (2\right )+36 \cos \left (2\right ) a_{1} {\mathrm e}^{2} \sin \left (2\right )+256 a_{1}-12 a_{1} \sin \left (2\right )^{3}+24 \cos \left (2\right )^{3} a_{1}-8 \,{\mathrm e}^{4} a_{0} \sin \left (2\right )-38 a_{0} {\mathrm e}^{2} \sin \left (2\right )^{2}-36 a_{1} {\mathrm e}^{2} \sin \left (2\right )^{2}-44 \cos \left (2\right )^{2} a_{0} {\mathrm e}^{2}-224 \cos \left (2\right ) a_{1} \sin \left (2\right )+48 a_{1} {\mathrm e}^{2} \sin \left (2\right )-96 \cos \left (2\right ) a_{0} {\mathrm e}^{2}-384 a_{1} \sin \left (2\right )+384 \cos \left (2\right ) a_{1}+4 a_{0} {\mathrm e}^{2} \sin \left (2\right )^{3}+6 a_{1} {\mathrm e}^{2} \sin \left (2\right )^{3}+144 a_{1} \sin \left (2\right )^{2}+176 \cos \left (2\right )^{2} a_{1}-6 \cos \left (2\right )^{3} a_{0} {\mathrm e}^{2}+26 \cos \left (2\right ) a_{1} \sin \left (2\right )^{2}-24 \cos \left (2\right )^{2} a_{1} \sin \left (2\right )+4 \,{\mathrm e}^{4} a_{0} \sin \left (2\right )^{2}+{\mathrm e}^{4} a_{1} \sin \left (2\right )^{2}+96 a_{0} {\mathrm e}^{2} \sin \left (2\right )}{120 \sin \left (2\right )^{4}} \] \(n=4\) gives \[ \frac {a_{2} \sin \left (2\right )}{12}-a_{3} \cos \left (2\right )-6 a_{4} \sin \left (2\right )+20 a_{5} \cos \left (2\right )+30 a_{6} \sin \left (2\right )+3 a_{3}+16 a_{4}+20 a_{5}-a_{4} {\mathrm e}^{2}-a_{3} {\mathrm e}^{2}-\frac {a_{2} {\mathrm e}^{2}}{2}-\frac {a_{1} {\mathrm e}^{2}}{6}-\frac {a_{0} {\mathrm e}^{2}}{24}=0 \] Which after substituting earlier equations, simplifies to \[ a_{6} = \frac {256 a_{0} {\mathrm e}^{2}+284 \cos \left (2\right ) a_{0} {\mathrm e}^{2} \sin \left (2\right )^{2}+224 \cos \left (2\right ) a_{1} {\mathrm e}^{2} \sin \left (2\right )^{2}-280 \cos \left (2\right )^{2} a_{0} {\mathrm e}^{2} \sin \left (2\right )-176 \cos \left (2\right )^{2} a_{1} {\mathrm e}^{2} \sin \left (2\right )+60 \cos \left (2\right ) {\mathrm e}^{4} a_{0} \sin \left (2\right )-800 \cos \left (2\right ) a_{0} {\mathrm e}^{2} \sin \left (2\right )-384 \cos \left (2\right ) a_{1} {\mathrm e}^{2} \sin \left (2\right )-22 \cos \left (2\right ) {\mathrm e}^{4} a_{0} \sin \left (2\right )^{2}-6 \cos \left (2\right ) {\mathrm e}^{4} a_{1} \sin \left (2\right )^{2}+18 \cos \left (2\right )^{2} {\mathrm e}^{4} a_{0} \sin \left (2\right )-1024 a_{1}+480 a_{1} \sin \left (2\right )^{3}-800 \cos \left (2\right )^{3} a_{1}+12 a_{0} {\mathrm e}^{2} \sin \left (2\right )^{4}+16 a_{1} {\mathrm e}^{2} \sin \left (2\right )^{4}+24 \cos \left (2\right )^{4} a_{0} {\mathrm e}^{2}+80 \cos \left (2\right ) a_{1} \sin \left (2\right )^{3}-136 \cos \left (2\right )^{2} a_{1} \sin \left (2\right )^{2}+48 \,{\mathrm e}^{4} a_{0} \sin \left (2\right )+488 a_{0} {\mathrm e}^{2} \sin \left (2\right )^{2}+384 a_{1} {\mathrm e}^{2} \sin \left (2\right )^{2}+560 \cos \left (2\right )^{2} a_{0} {\mathrm e}^{2}+3200 \cos \left (2\right ) a_{1} \sin \left (2\right )-24 \cos \left (2\right ) a_{0} {\mathrm e}^{2} \sin \left (2\right )^{3}-32 \cos \left (2\right ) a_{1} {\mathrm e}^{2} \sin \left (2\right )^{3}+40 \cos \left (2\right )^{2} a_{0} {\mathrm e}^{2} \sin \left (2\right )^{2}+24 \cos \left (2\right )^{2} a_{1} {\mathrm e}^{2} \sin \left (2\right )^{2}-24 \cos \left (2\right )^{3} a_{0} {\mathrm e}^{2} \sin \left (2\right )-24 \cos \left (2\right )^{3} a_{1} {\mathrm e}^{2} \sin \left (2\right )-256 a_{1} {\mathrm e}^{2} \sin \left (2\right )+640 \cos \left (2\right ) a_{0} {\mathrm e}^{2}+96 \cos \left (2\right )^{3} a_{1} \sin \left (2\right )+18 \,{\mathrm e}^{4} a_{0} \sin \left (2\right )^{3}+6 \,{\mathrm e}^{4} a_{1} \sin \left (2\right )^{3}+{\mathrm e}^{6} a_{0} \sin \left (2\right )^{2}+2560 a_{1} \sin \left (2\right )-2560 \cos \left (2\right ) a_{1}-132 a_{0} {\mathrm e}^{2} \sin \left (2\right )^{3}-162 a_{1} {\mathrm e}^{2} \sin \left (2\right )^{3}-1920 a_{1} \sin \left (2\right )^{2}-2240 \cos \left (2\right )^{2} a_{1}-32 a_{1} \sin \left (2\right )^{4}-96 \cos \left (2\right )^{4} a_{1}+200 \cos \left (2\right )^{3} a_{0} {\mathrm e}^{2}-1080 \cos \left (2\right ) a_{1} \sin \left (2\right )^{2}+1120 \cos \left (2\right )^{2} a_{1} \sin \left (2\right )-60 \,{\mathrm e}^{4} a_{0} \sin \left (2\right )^{2}-12 \,{\mathrm e}^{4} a_{1} \sin \left (2\right )^{2}-640 a_{0} {\mathrm e}^{2} \sin \left (2\right )}{720 \sin \left (2\right )^{5}} \] \(n=5\) gives \[ \frac {a_{2} \cos \left (2\right )}{60}+\frac {a_{3} \sin \left (2\right )}{4}-2 a_{4} \cos \left (2\right )-10 a_{5} \sin \left (2\right )+30 a_{6} \cos \left (2\right )+42 a_{7} \sin \left (2\right )+4 a_{4}+20 a_{5}+24 a_{6}-a_{5} {\mathrm e}^{2}-a_{4} {\mathrm e}^{2}-\frac {a_{3} {\mathrm e}^{2}}{2}-\frac {a_{2} {\mathrm e}^{2}}{6}-\frac {a_{1} {\mathrm e}^{2}}{24}-\frac {a_{0} {\mathrm e}^{2}}{120}=0 \] 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 ) \sin \left (2\right )}{720}+\frac {\left (n -3\right ) a_{n -3} \left (n -4\right ) \cos \left (2\right )}{120}+\frac {\left (n -2\right ) a_{n -2} \left (n -3\right ) \sin \left (2\right )}{24}-\frac {\left (n -1\right ) a_{n -1} \left (n -2\right ) \cos \left (2\right )}{6}-\frac {n a_{n} \left (n -1\right ) \sin \left (2\right )}{2}+\left (1+n \right ) a_{1+n} n \cos \left (2\right )+\left (n +2\right ) a_{n +2} \left (1+n \right ) \sin \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 {-2880 a_{1+n}+720 a_{n -1}+720 a_{n -1} {\mathrm e}^{2}+a_{n -6} {\mathrm e}^{2}+30 a_{n -4} {\mathrm e}^{2}+120 a_{n -3} {\mathrm e}^{2}+360 a_{n -2} {\mathrm e}^{2}+6 a_{n -5} {\mathrm e}^{2}-720 a_{n -1} n -2880 a_{1+n} n +20 a_{n -4} \sin \left (2\right )-180 a_{n -2} \sin \left (2\right )-72 a_{n -3} \cos \left (2\right )+240 a_{n -1} \cos \left (2\right )+360 n^{2} a_{n} \sin \left (2\right )+\sin \left (2\right ) n^{2} a_{n -4}-30 \sin \left (2\right ) n^{2} a_{n -2}-720 a_{1+n} n^{2} \cos \left (2\right )-6 \cos \left (2\right ) n^{2} a_{n -3}+120 \cos \left (2\right ) n^{2} a_{n -1}-360 n a_{n} \sin \left (2\right )-9 \sin \left (2\right ) n a_{n -4}+150 \sin \left (2\right ) n a_{n -2}-720 a_{1+n} n \cos \left (2\right )+42 \cos \left (2\right ) n a_{n -3}-360 \cos \left (2\right ) n a_{n -1}-2880 n a_{n}+720 a_{n} {\mathrm e}^{2}}{720 \sin \left (2\right ) \left (n^{2}+3 n +2\right )} \\ &= \frac {\left (360 \sin \left (2\right ) n^{2}-360 \sin \left (2\right ) n +720 \,{\mathrm e}^{2}-2880 n \right ) a_{n}}{720 \sin \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\left (-720 \cos \left (2\right ) n^{2}-720 \cos \left (2\right ) n -2880 n -2880\right ) a_{1+n}}{720 \sin \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {{\mathrm e}^{2} a_{n -6}}{720 \sin \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {{\mathrm e}^{2} a_{n -5}}{120 \sin \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\left (\sin \left (2\right ) n^{2}-9 \sin \left (2\right ) n +20 \sin \left (2\right )+30 \,{\mathrm e}^{2}\right ) a_{n -4}}{720 \sin \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\left (-6 \cos \left (2\right ) n^{2}+42 \cos \left (2\right ) n -72 \cos \left (2\right )+120 \,{\mathrm e}^{2}\right ) a_{n -3}}{720 \sin \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\left (-30 \sin \left (2\right ) n^{2}+150 \sin \left (2\right ) n -180 \sin \left (2\right )+360 \,{\mathrm e}^{2}\right ) a_{n -2}}{720 \sin \left (2\right ) \left (n^{2}+3 n +2\right )}+\frac {\left (120 \cos \left (2\right ) n^{2}-360 \cos \left (2\right ) n +240 \cos \left (2\right )+720 \,{\mathrm e}^{2}-720 n +720\right ) a_{n -1}}{720 \sin \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 \sin \left (2\right )}+\frac {\left (a_{0} {\mathrm e}^{2} \sin \left (2\right )+a_{1} {\mathrm e}^{2} \sin \left (2\right )-\cos \left (2\right ) a_{0} {\mathrm e}^{2}-4 a_{1} \sin \left (2\right )+4 \cos \left (2\right ) a_{1}-4 a_{0} {\mathrm e}^{2}+16 a_{1}\right ) t^{3}}{6 \sin \left (2\right )^{2}}+\frac {\left (2 a_{0} {\mathrm e}^{2} \sin \left (2\right )^{2}+2 a_{1} {\mathrm e}^{2} \sin \left (2\right )^{2}-2 \cos \left (2\right ) a_{0} {\mathrm e}^{2} \sin \left (2\right )-2 \cos \left (2\right ) a_{1} {\mathrm e}^{2} \sin \left (2\right )+{\mathrm e}^{4} a_{0} \sin \left (2\right )+2 \cos \left (2\right )^{2} a_{0} {\mathrm e}^{2}-6 a_{1} \sin \left (2\right )^{2}+8 \cos \left (2\right ) a_{1} \sin \left (2\right )-12 a_{0} {\mathrm e}^{2} \sin \left (2\right )-8 a_{1} {\mathrm e}^{2} \sin \left (2\right )-8 \cos \left (2\right )^{2} a_{1}+12 \cos \left (2\right ) a_{0} {\mathrm e}^{2}+48 a_{1} \sin \left (2\right )-48 \cos \left (2\right ) a_{1}+16 a_{0} {\mathrm e}^{2}-64 a_{1}\right ) t^{4}}{24 \sin \left (2\right )^{3}}+\frac {\left (-64 a_{0} {\mathrm e}^{2}-8 \cos \left (2\right ) a_{0} {\mathrm e}^{2} \sin \left (2\right )^{2}-6 \cos \left (2\right ) a_{1} {\mathrm e}^{2} \sin \left (2\right )^{2}+6 \cos \left (2\right )^{2} a_{0} {\mathrm e}^{2} \sin \left (2\right )+6 \cos \left (2\right )^{2} a_{1} {\mathrm e}^{2} \sin \left (2\right )-4 \cos \left (2\right ) {\mathrm e}^{4} a_{0} \sin \left (2\right )+56 \cos \left (2\right ) a_{0} {\mathrm e}^{2} \sin \left (2\right )+36 \cos \left (2\right ) a_{1} {\mathrm e}^{2} \sin \left (2\right )+256 a_{1}-12 a_{1} \sin \left (2\right )^{3}+24 \cos \left (2\right )^{3} a_{1}-8 \,{\mathrm e}^{4} a_{0} \sin \left (2\right )-38 a_{0} {\mathrm e}^{2} \sin \left (2\right )^{2}-36 a_{1} {\mathrm e}^{2} \sin \left (2\right )^{2}-44 \cos \left (2\right )^{2} a_{0} {\mathrm e}^{2}-224 \cos \left (2\right ) a_{1} \sin \left (2\right )+48 a_{1} {\mathrm e}^{2} \sin \left (2\right )-96 \cos \left (2\right ) a_{0} {\mathrm e}^{2}-384 a_{1} \sin \left (2\right )+384 \cos \left (2\right ) a_{1}+4 a_{0} {\mathrm e}^{2} \sin \left (2\right )^{3}+6 a_{1} {\mathrm e}^{2} \sin \left (2\right )^{3}+144 a_{1} \sin \left (2\right )^{2}+176 \cos \left (2\right )^{2} a_{1}-6 \cos \left (2\right )^{3} a_{0} {\mathrm e}^{2}+26 \cos \left (2\right ) a_{1} \sin \left (2\right )^{2}-24 \cos \left (2\right )^{2} a_{1} \sin \left (2\right )+4 \,{\mathrm e}^{4} a_{0} \sin \left (2\right )^{2}+{\mathrm e}^{4} a_{1} \sin \left (2\right )^{2}+96 a_{0} {\mathrm e}^{2} \sin \left (2\right )\right ) t^{5}}{120 \sin \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 \sin \left (2\right )}+\frac {\left (\sin \left (2\right ) {\mathrm e}^{2}-{\mathrm e}^{2} \cos \left (2\right )-4 \,{\mathrm e}^{2}\right ) t^{3}}{6 \sin \left (2\right )^{2}}+\frac {\left (2 \sin \left (2\right )^{2} {\mathrm e}^{2}-2 \sin \left (2\right ) \cos \left (2\right ) {\mathrm e}^{2}+{\mathrm e}^{4} \sin \left (2\right )+2 \cos \left (2\right )^{2} {\mathrm e}^{2}-12 \sin \left (2\right ) {\mathrm e}^{2}+12 \,{\mathrm e}^{2} \cos \left (2\right )+16 \,{\mathrm e}^{2}\right ) t^{4}}{24 \sin \left (2\right )^{3}}+\frac {\left (4 \sin \left (2\right )^{3} {\mathrm e}^{2}-8 \sin \left (2\right )^{2} \cos \left (2\right ) {\mathrm e}^{2}+4 \sin \left (2\right )^{2} {\mathrm e}^{4}+6 \sin \left (2\right ) \cos \left (2\right )^{2} {\mathrm e}^{2}-4 \sin \left (2\right ) \cos \left (2\right ) {\mathrm e}^{4}-6 \cos \left (2\right )^{3} {\mathrm e}^{2}-38 \sin \left (2\right )^{2} {\mathrm e}^{2}+56 \sin \left (2\right ) \cos \left (2\right ) {\mathrm e}^{2}-8 \,{\mathrm e}^{4} \sin \left (2\right )-44 \cos \left (2\right )^{2} {\mathrm e}^{2}+96 \sin \left (2\right ) {\mathrm e}^{2}-96 \,{\mathrm e}^{2} \cos \left (2\right )-64 \,{\mathrm e}^{2}\right ) t^{5}}{120 \sin \left (2\right )^{4}}\right ) a_{0}+\left (t -\frac {2 t^{2}}{\sin \left (2\right )}+\frac {\left (\sin \left (2\right ) {\mathrm e}^{2}-4 \sin \left (2\right )+4 \cos \left (2\right )+16\right ) t^{3}}{6 \sin \left (2\right )^{2}}+\frac {\left (2 \sin \left (2\right )^{2} {\mathrm e}^{2}-2 \sin \left (2\right ) \cos \left (2\right ) {\mathrm e}^{2}-6 \sin \left (2\right )^{2}+8 \cos \left (2\right ) \sin \left (2\right )-8 \sin \left (2\right ) {\mathrm e}^{2}-8 \cos \left (2\right )^{2}+48 \sin \left (2\right )-48 \cos \left (2\right )-64\right ) t^{4}}{24 \sin \left (2\right )^{3}}+\frac {\left (6 \sin \left (2\right )^{3} {\mathrm e}^{2}-6 \sin \left (2\right )^{2} \cos \left (2\right ) {\mathrm e}^{2}+\sin \left (2\right )^{2} {\mathrm e}^{4}+6 \sin \left (2\right ) \cos \left (2\right )^{2} {\mathrm e}^{2}-12 \sin \left (2\right )^{3}+26 \cos \left (2\right ) \sin \left (2\right )^{2}-36 \sin \left (2\right )^{2} {\mathrm e}^{2}-24 \cos \left (2\right )^{2} \sin \left (2\right )+36 \sin \left (2\right ) \cos \left (2\right ) {\mathrm e}^{2}+24 \cos \left (2\right )^{3}+144 \sin \left (2\right )^{2}-224 \cos \left (2\right ) \sin \left (2\right )+48 \sin \left (2\right ) {\mathrm e}^{2}+176 \cos \left (2\right )^{2}-384 \sin \left (2\right )+384 \cos \left (2\right )+256\right ) t^{5}}{120 \sin \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 \sin \left (2\right )}+\frac {\left (\sin \left (2\right ) {\mathrm e}^{2}-{\mathrm e}^{2} \cos \left (2\right )-4 \,{\mathrm e}^{2}\right ) t^{3}}{6 \sin \left (2\right )^{2}}+\frac {\left (2 \sin \left (2\right )^{2} {\mathrm e}^{2}-2 \sin \left (2\right ) \cos \left (2\right ) {\mathrm e}^{2}+{\mathrm e}^{4} \sin \left (2\right )+2 \cos \left (2\right )^{2} {\mathrm e}^{2}-12 \sin \left (2\right ) {\mathrm e}^{2}+12 \,{\mathrm e}^{2} \cos \left (2\right )+16 \,{\mathrm e}^{2}\right ) t^{4}}{24 \sin \left (2\right )^{3}}+\frac {\left (4 \sin \left (2\right )^{3} {\mathrm e}^{2}-8 \sin \left (2\right )^{2} \cos \left (2\right ) {\mathrm e}^{2}+4 \sin \left (2\right )^{2} {\mathrm e}^{4}+6 \sin \left (2\right ) \cos \left (2\right )^{2} {\mathrm e}^{2}-4 \sin \left (2\right ) \cos \left (2\right ) {\mathrm e}^{4}-6 \cos \left (2\right )^{3} {\mathrm e}^{2}-38 \sin \left (2\right )^{2} {\mathrm e}^{2}+56 \sin \left (2\right ) \cos \left (2\right ) {\mathrm e}^{2}-8 \,{\mathrm e}^{4} \sin \left (2\right )-44 \cos \left (2\right )^{2} {\mathrm e}^{2}+96 \sin \left (2\right ) {\mathrm e}^{2}-96 \,{\mathrm e}^{2} \cos \left (2\right )-64 \,{\mathrm e}^{2}\right ) t^{5}}{120 \sin \left (2\right )^{4}}\right ) c_{1} +\left (t -\frac {2 t^{2}}{\sin \left (2\right )}+\frac {\left (\sin \left (2\right ) {\mathrm e}^{2}-4 \sin \left (2\right )+4 \cos \left (2\right )+16\right ) t^{3}}{6 \sin \left (2\right )^{2}}+\frac {\left (2 \sin \left (2\right )^{2} {\mathrm e}^{2}-2 \sin \left (2\right ) \cos \left (2\right ) {\mathrm e}^{2}-6 \sin \left (2\right )^{2}+8 \cos \left (2\right ) \sin \left (2\right )-8 \sin \left (2\right ) {\mathrm e}^{2}-8 \cos \left (2\right )^{2}+48 \sin \left (2\right )-48 \cos \left (2\right )-64\right ) t^{4}}{24 \sin \left (2\right )^{3}}+\frac {\left (6 \sin \left (2\right )^{3} {\mathrm e}^{2}-6 \sin \left (2\right )^{2} \cos \left (2\right ) {\mathrm e}^{2}+\sin \left (2\right )^{2} {\mathrm e}^{4}+6 \sin \left (2\right ) \cos \left (2\right )^{2} {\mathrm e}^{2}-12 \sin \left (2\right )^{3}+26 \cos \left (2\right ) \sin \left (2\right )^{2}-36 \sin \left (2\right )^{2} {\mathrm e}^{2}-24 \cos \left (2\right )^{2} \sin \left (2\right )+36 \sin \left (2\right ) \cos \left (2\right ) {\mathrm e}^{2}+24 \cos \left (2\right )^{3}+144 \sin \left (2\right )^{2}-224 \cos \left (2\right ) \sin \left (2\right )+48 \sin \left (2\right ) {\mathrm e}^{2}+176 \cos \left (2\right )^{2}-384 \sin \left (2\right )+384 \cos \left (2\right )+256\right ) t^{5}}{120 \sin \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: 509

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

\[ y \left (x \right ) = \left (1+\frac {\csc \left (2\right ) {\mathrm e}^{2} \left (x -2\right )^{2}}{2}-\frac {{\mathrm e}^{2} \left (4+\cos \left (2\right )-\sin \left (2\right )\right ) \csc \left (2\right )^{2} \left (x -2\right )^{3}}{6}+\frac {\csc \left (2\right )^{3} \left (\left (\frac {3}{2}-\frac {\sin \left (4\right )}{12}-\sin \left (2\right )+\cos \left (2\right )\right ) {\mathrm e}^{2}+\frac {{\mathrm e}^{4} \sin \left (2\right )}{12}\right ) \left (x -2\right )^{4}}{2}+\frac {\left (\left (-210+56 \sin \left (4\right )+\sin \left (6\right )+201 \sin \left (2\right )+\cos \left (6\right )-205 \cos \left (2\right )-6 \cos \left (4\right )\right ) {\mathrm e}^{2}-4 \,{\mathrm e}^{4} \left (-1+\sin \left (4\right )+\cos \left (4\right )+4 \sin \left (2\right )\right )\right ) \csc \left (2\right )^{4} \left (x -2\right )^{5}}{240}\right ) y \left (2\right )+\left (x -2-2 \csc \left (2\right ) \left (x -2\right )^{2}-\frac {\left (-{\mathrm e}^{2} \sin \left (2\right )-4 \cos \left (2\right )+4 \sin \left (2\right )-16\right ) \csc \left (2\right )^{2} \left (x -2\right )^{3}}{6}+\frac {\csc \left (2\right )^{3} \left (\left (-\frac {\cos \left (4\right )}{12}-\frac {2 \sin \left (2\right )}{3}-\frac {\sin \left (4\right )}{12}+\frac {1}{12}\right ) {\mathrm e}^{2}-4 \cos \left (2\right )-\frac {\cos \left (4\right )}{12}+4 \sin \left (2\right )+\frac {\sin \left (4\right )}{3}-\frac {71}{12}\right ) \left (x -2\right )^{4}}{2}+\frac {\left (\left (\left (-12 \cos \left (2\right )-72\right ) \sin \left (2\right )^{2}+108 \sin \left (2\right )+36 \sin \left (4\right )\right ) {\mathrm e}^{2}+2 \sin \left (2\right )^{2} {\mathrm e}^{4}-6 \sin \left (6\right )+817 \cos \left (2\right )+32 \cos \left (4\right )-\cos \left (6\right )-798 \sin \left (2\right )-224 \sin \left (4\right )+832\right ) \csc \left (2\right )^{4} \left (x -2\right )^{5}}{240}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 953

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

\[ y(x)\to c_2 \left (-\frac {1}{60} \left (6 \csc (2)-13 \cot (2) \csc (2)+12 \cot ^2(2) \csc (2)-12 \cot ^3(2) \csc (2)\right ) (x-2)^5-\frac {1}{20} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)-e^2 \cot ^2(2) \csc (2)\right ) (x-2)^5+\frac {4}{15} \csc (2) \left (3 \csc (2)-4 \cot (2) \csc (2)+4 \cot ^2(2) \csc (2)\right ) (x-2)^5+\frac {1}{6} \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {1}{40} (4 \csc (2)-4 \cot (2) \csc (2)) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5+\frac {2}{5} \csc ^2(2) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5+\frac {1}{40} e^2 \csc (2) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5-\frac {2}{5} \csc ^2(2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^5-\frac {1}{120} e^2 \csc (2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^5+\frac {32}{15} \csc ^4(2) (x-2)^5+\frac {2}{5} e^2 \csc ^3(2) (x-2)^5+\frac {1}{120} e^4 \csc ^2(2) (x-2)^5-\frac {1}{12} \left (3 \csc (2)-4 \cot (2) \csc (2)+4 \cot ^2(2) \csc (2)\right ) (x-2)^4-\frac {1}{12} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^4+\frac {1}{2} \csc (2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^4-\frac {8}{3} \csc ^3(2) (x-2)^4-\frac {1}{3} e^2 \csc ^2(2) (x-2)^4-\frac {1}{6} (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^3+\frac {8}{3} \csc ^2(2) (x-2)^3+\frac {1}{6} e^2 \csc (2) (x-2)^3-2 \csc (2) (x-2)^2+x-2\right )+c_1 \left (-\frac {1}{60} \left (-2 e^2 \csc (2)+4 e^2 \cot (2) \csc (2)-3 e^2 \cot ^2(2) \csc (2)+3 e^2 \cot ^3(2) \csc (2)\right ) (x-2)^5+\frac {1}{15} \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)-e^2 \cot ^2(2) \csc (2)\right ) (x-2)^5-\frac {1}{20} e^2 \csc (2) \left (3 \csc (2)-4 \cot (2) \csc (2)+4 \cot ^2(2) \csc (2)\right ) (x-2)^5-\frac {1}{40} (-4 \csc (2)+4 \cot (2) \csc (2)) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {2}{15} \csc ^2(2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {1}{30} e^2 \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {1}{10} e^2 \csc ^2(2) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5+\frac {1}{15} e^2 \csc ^2(2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^5-\frac {8}{15} e^2 \csc ^4(2) (x-2)^5-\frac {1}{15} e^4 \csc ^3(2) (x-2)^5-\frac {1}{12} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)-e^2 \cot ^2(2) \csc (2)\right ) (x-2)^4+\frac {1}{6} \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^4-\frac {1}{12} e^2 \csc (2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^4+\frac {2}{3} e^2 \csc ^3(2) (x-2)^4+\frac {1}{24} e^4 \csc ^2(2) (x-2)^4-\frac {1}{6} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^3-\frac {2}{3} e^2 \csc ^2(2) (x-2)^3+\frac {1}{2} e^2 \csc (2) (x-2)^2+1\right ) \]