Internal problem ID [5558]
Internal file name [OUTPUT/4806_Sunday_June_05_2022_03_06_16_PM_27990159/index.tex]
Book: A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications.
Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section: Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page
239
Problem number: 3.
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]]
\[ \boxed {\left (x^{2}-9\right )^{2} y^{\prime \prime }+\left (x +3\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
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 {y^{\prime } x +3 y^{\prime }+2 y}{x^{4}-18 x^{2}+81}\\ F_1 &= \frac {d F_0}{dx} \\ &= \frac {\partial F_{0}}{\partial x}+ \frac {\partial F_{0}}{\partial y} y^{\prime }+ \frac {\partial F_{0}}{\partial y^{\prime }} F_0 \\ &= \frac {\left (x^{3}+9 x^{2}-8 x -78\right ) y^{\prime }+8 \left (x^{2}-3 x +\frac {1}{4}\right ) y}{\left (x -3\right )^{4} \left (x +3\right )^{3}}\\ F_2 &= \frac {d F_1}{dx} \\ &= \frac {\partial F_{1}}{\partial x}+ \frac {\partial F_{1}}{\partial y} y^{\prime }+ \frac {\partial F_{1}}{\partial y^{\prime }} F_1 \\ &= \frac {\left (4 x^{5}-84 x^{4}+139 x^{3}+843 x^{2}-1576 x -786\right ) y^{\prime }-40 y \left (x^{4}-6 x^{3}+\frac {56}{5} x^{2}-\frac {57}{5} x +\frac {289}{20}\right )}{\left (x +3\right )^{4} \left (x -3\right )^{6}}\\ F_3 &= \frac {d F_2}{dx} \\ &= \frac {\partial F_{2}}{\partial x}+ \frac {\partial F_{2}}{\partial y} y^{\prime }+ \frac {\partial F_{2}}{\partial y^{\prime }} F_2 \\ &= \frac {\left (-60 x^{7}+900 x^{6}-2901 x^{5}-3963 x^{4}+27462 x^{3}-29286 x^{2}+31564 x -71520\right ) y^{\prime }+240 y \left (x^{6}-9 x^{5}+\frac {329}{10} x^{4}-\frac {769}{10} x^{3}+\frac {5657}{40} x^{2}-\frac {2757}{20} x +\frac {152}{15}\right )}{\left (x -3\right )^{8} \left (x +3\right )^{5}}\\ F_4 &= \frac {d F_3}{dx} \\ &= \frac {\partial F_{3}}{\partial x}+ \frac {\partial F_{3}}{\partial y} y^{\prime }+ \frac {\partial F_{3}}{\partial y^{\prime }} F_3 \\ &= \frac {\left (600 x^{9}-9720 x^{8}+48444 x^{7}-60612 x^{6}-185109 x^{5}+791229 x^{4}-2239204 x^{3}+4979016 x^{2}-4105612 x -941628\right ) y^{\prime }-1680 \left (x^{8}-12 x^{7}+\frac {2288}{35} x^{6}-\frac {8381}{35} x^{5}+\frac {1323}{2} x^{4}-\frac {84811}{70} x^{3}+\frac {1041253}{840} x^{2}-\frac {21223}{28} x +\frac {194981}{420}\right ) y}{\left (x +3\right )^{6} \left (x -3\right )^{10}} \end {align*}
And so on. Evaluating all the above at initial conditions \(x = 0\) 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 y \left (0\right )}{81}-\frac {y^{\prime }\left (0\right )}{27}\\ F_1 &= \frac {2 y \left (0\right )}{2187}-\frac {26 y^{\prime }\left (0\right )}{729}\\ F_2 &= -\frac {578 y \left (0\right )}{59049}-\frac {262 y^{\prime }\left (0\right )}{19683}\\ F_3 &= \frac {2432 y \left (0\right )}{1594323}-\frac {23840 y^{\prime }\left (0\right )}{531441}\\ F_4 &= -\frac {779924 y \left (0\right )}{43046721}-\frac {313876 y^{\prime }\left (0\right )}{14348907} \end {align*}
Substituting all the above in (7) and simplifying gives the solution as \[ y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}-\frac {194981}{7748409780} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}-\frac {78469}{2582803260} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Since the expansion point \(x = 0\) is an ordinary, we can also solve this using standard power series The ode is normalized to be \[ y^{\prime \prime } \left (x^{4}-18 x^{2}+81\right )+\left (x +3\right ) y^{\prime }+2 y = 0 \] Let the solution be represented as power series of the form \[ y = \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n} \] Then \begin {align*} y^{\prime } &= \moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\\ y^{\prime \prime } &= \moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} x^{n -2} \end {align*}
Substituting the above back into the ode gives \begin {align*} \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} x^{n -2}\right ) \left (x^{4}-18 x^{2}+81\right )+\left (x +3\right ) \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\right )+2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\right ) = 0\tag {1} \end {align*}
Which simplifies to \begin{equation} \tag{2} \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \,x^{n +2} a_{n} \left (n -1\right )\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-18 x^{n} a_{n} n \left (n -1\right )\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}81 n \left (n -1\right ) a_{n} x^{n -2}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}3 n a_{n} x^{n -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 a_{n} x^{n}\right ) = 0 \end{equation} The next step is to make all powers of \(x\) be \(n\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =2}{\sum }}n \,x^{n +2} a_{n} \left (n -1\right ) &= \moverset {\infty }{\munderset {n =4}{\sum }}\left (n -2\right ) a_{n -2} \left (n -3\right ) x^{n} \\ \moverset {\infty }{\munderset {n =2}{\sum }}81 n \left (n -1\right ) a_{n} x^{n -2} &= \moverset {\infty }{\munderset {n =0}{\sum }}81 \left (n +2\right ) a_{n +2} \left (n +1\right ) x^{n} \\ \moverset {\infty }{\munderset {n =1}{\sum }}3 n a_{n} x^{n -1} &= \moverset {\infty }{\munderset {n =0}{\sum }}3 \left (n +1\right ) a_{n +1} x^{n} \\ \end{align*} Substituting all the above in Eq (2) gives the following equation where now all powers of \(x\) are the same and equal to \(n\). \begin{equation} \tag{3} \left (\moverset {\infty }{\munderset {n =4}{\sum }}\left (n -2\right ) a_{n -2} \left (n -3\right ) x^{n}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-18 x^{n} a_{n} n \left (n -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}81 \left (n +2\right ) a_{n +2} \left (n +1\right ) x^{n}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}3 \left (n +1\right ) a_{n +1} x^{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 a_{n} x^{n}\right ) = 0 \end{equation} \(n=0\) gives \[ 162 a_{2}+3 a_{1}+2 a_{0}=0 \] \[ a_{2} = -\frac {a_{0}}{81}-\frac {a_{1}}{54} \] \(n=1\) gives \[ 486 a_{3}+3 a_{1}+6 a_{2}=0 \] Which after substituting earlier equations, simplifies to \[ a_{3} = \frac {a_{0}}{6561}-\frac {13 a_{1}}{2187} \] \(n=2\) gives \[ -32 a_{2}+972 a_{4}+9 a_{3} = 0 \] Which after substituting earlier equations, simplifies to \[ \frac {289 a_{0}}{729}+\frac {131 a_{1}}{243}+972 a_{4} = 0 \] Or \[ a_{4} = -\frac {289 a_{0}}{708588}-\frac {131 a_{1}}{236196} \] \(n=3\) gives \[ -103 a_{3}+1620 a_{5}+12 a_{4} = 0 \] Which after substituting earlier equations, simplifies to \[ -\frac {1216 a_{0}}{59049}+\frac {11920 a_{1}}{19683}+1620 a_{5} = 0 \] Or \[ a_{5} = \frac {304 a_{0}}{23914845}-\frac {596 a_{1}}{1594323} \] For \(4\le n\), the recurrence equation is \begin{equation} \tag{4} \left (n -2\right ) a_{n -2} \left (n -3\right )-18 n a_{n} \left (n -1\right )+81 \left (n +2\right ) a_{n +2} \left (n +1\right )+n a_{n}+3 \left (n +1\right ) a_{n +1}+2 a_{n} = 0 \end{equation} Solving for \(a_{n +2}\), gives \begin{align*} \tag{5} a_{n +2}&= \frac {18 n^{2} a_{n}-n^{2} a_{n -2}-19 n a_{n}+5 n a_{n -2}-3 n a_{n +1}-2 a_{n}-6 a_{n -2}-3 a_{n +1}}{81 \left (n +2\right ) \left (n +1\right )} \\ &= \frac {\left (18 n^{2}-19 n -2\right ) a_{n}}{81 \left (n +2\right ) \left (n +1\right )}+\frac {\left (-n^{2}+5 n -6\right ) a_{n -2}}{81 \left (n +2\right ) \left (n +1\right )}+\frac {\left (-3 n -3\right ) a_{n +1}}{81 \left (n +2\right ) \left (n +1\right )} \\ \end{align*} For \(n = 4\) the recurrence equation gives \[ 2 a_{2}-210 a_{4}+2430 a_{6}+15 a_{5} = 0 \] Which after substituting the earlier terms found becomes \[ a_{6} = -\frac {194981 a_{0}}{7748409780}-\frac {78469 a_{1}}{2582803260} \] For \(n = 5\) the recurrence equation gives \[ 6 a_{3}-353 a_{5}+3402 a_{7}+18 a_{6} = 0 \] Which after substituting the earlier terms found becomes \[ a_{7} = \frac {1732937 a_{0}}{1464449448420}-\frac {13738871 a_{1}}{488149816140} \] And so on. Therefore the solution is \begin {align*} y &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\\ &= a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0} + \dots \end {align*}
Substituting the values for \(a_{n}\) found above, the solution becomes \[ y = a_{0}+a_{1} x +\left (-\frac {a_{0}}{81}-\frac {a_{1}}{54}\right ) x^{2}+\left (\frac {a_{0}}{6561}-\frac {13 a_{1}}{2187}\right ) x^{3}+\left (-\frac {289 a_{0}}{708588}-\frac {131 a_{1}}{236196}\right ) x^{4}+\left (\frac {304 a_{0}}{23914845}-\frac {596 a_{1}}{1594323}\right ) x^{5}+\dots \] Collecting terms, the solution becomes \begin{equation} \tag{3} y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}\right ) a_{0}+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}\right ) a_{1}+O\left (x^{6}\right ) \end{equation} At \(x = 0\) the solution above becomes \[ y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}\right ) c_{1} +\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}-\frac {194981}{7748409780} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}-\frac {78469}{2582803260} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \\ \tag{2} y &= \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}\right ) c_{1} +\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \\ \end{align*}
Verification of solutions
\[ y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}-\frac {194981}{7748409780} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}-\frac {78469}{2582803260} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK.
\[ y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}\right ) c_{1} +\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime } \left (x^{4}-18 x^{2}+81\right )+\left (x +3\right ) y^{\prime }+2 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=-\frac {2 y}{x^{4}-18 x^{2}+81}-\frac {y^{\prime }}{x^{3}-3 x^{2}-9 x +27} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime }+\frac {y^{\prime }}{x^{3}-3 x^{2}-9 x +27}+\frac {2 y}{x^{4}-18 x^{2}+81}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {1}{x^{3}-3 x^{2}-9 x +27}, P_{3}\left (x \right )=\frac {2}{x^{4}-18 x^{2}+81}\right ] \\ {} & \circ & \left (x +3\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-3 \\ {} & {} & \left (\left (x +3\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-3}}}=\frac {1}{36} \\ {} & \circ & \left (x +3\right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-3 \\ {} & {} & \left (\left (x +3\right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-3}}}=\frac {1}{18} \\ {} & \circ & x =-3\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=-3 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & y^{\prime \prime } \left (x^{3}-3 x^{2}-9 x +27\right ) \left (x^{4}-18 x^{2}+81\right )+y^{\prime } \left (x^{4}-18 x^{2}+81\right )+\left (2 x^{3}-6 x^{2}-18 x +54\right ) y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u -3\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (u^{7}-24 u^{6}+216 u^{5}-864 u^{4}+1296 u^{3}\right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (u^{4}-12 u^{3}+36 u^{2}\right ) \left (\frac {d}{d u}y \left (u \right )\right )+\left (2 u^{3}-24 u^{2}+72 u \right ) y \left (u \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \left (u \right ) \\ {} & {} & y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot y \left (u \right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..3 \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =2..4 \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) u^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =3..7 \\ {} & {} & u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) u^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) u^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & 36 a_{0} \left (36 r^{2}-35 r +2\right ) u^{1+r}+\left (36 a_{1} \left (36 r^{2}+37 r +3\right )-12 a_{0} \left (72 r^{2}-71 r +2\right )\right ) u^{2+r}+\left (36 a_{2} \left (36 r^{2}+109 r +76\right )-12 a_{1} \left (72 r^{2}+73 r +3\right )+a_{0} \left (216 r^{2}-215 r +2\right )\right ) u^{3+r}+\left (36 a_{3} \left (36 r^{2}+181 r +221\right )-12 a_{2} \left (72 r^{2}+217 r +148\right )+a_{1} \left (216 r^{2}+217 r +3\right )-24 a_{0} r \left (-1+r \right )\right ) u^{4+r}+\left (\moverset {\infty }{\munderset {k =5}{\sum }}\left (36 a_{k -1} \left (36 \left (k -1\right )^{2}+72 \left (k -1\right ) r +36 r^{2}-35 k +37-35 r \right )-12 a_{k -2} \left (72 \left (k -2\right )^{2}+144 \left (k -2\right ) r +72 r^{2}-71 k +144-71 r \right )+a_{k -3} \left (216 \left (k -3\right )^{2}+432 \left (k -3\right ) r +216 r^{2}-215 k +647-215 r \right )-24 a_{k -4} \left (k -4+r \right ) \left (k -5+r \right )+a_{k -5} \left (k -5+r \right ) \left (k -6+r \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 1296 r^{2}-1260 r +72=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{\frac {35}{72}-\frac {\sqrt {937}}{72}, \frac {35}{72}+\frac {\sqrt {937}}{72}\right \} \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} u \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [36 a_{1} \left (36 r^{2}+37 r +3\right )-12 a_{0} \left (72 r^{2}-71 r +2\right )=0, 36 a_{2} \left (36 r^{2}+109 r +76\right )-12 a_{1} \left (72 r^{2}+73 r +3\right )+a_{0} \left (216 r^{2}-215 r +2\right )=0, 36 a_{3} \left (36 r^{2}+181 r +221\right )-12 a_{2} \left (72 r^{2}+217 r +148\right )+a_{1} \left (216 r^{2}+217 r +3\right )-24 a_{0} r \left (-1+r \right )=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=\frac {a_{0} \left (72 r^{2}-71 r +2\right )}{3 \left (36 r^{2}+37 r +3\right )}, a_{2}=\frac {a_{0} \left (4320 r^{4}+108 r^{3}-4019 r^{2}+101 r +6\right )}{12 \left (1296 r^{4}+5256 r^{3}+6877 r^{2}+3139 r +228\right )}, a_{3}=\frac {a_{0} \left (233280 r^{6}+707616 r^{5}+277488 r^{4}-618425 r^{3}-414801 r^{2}+7442 r +1104\right )}{54 \left (46656 r^{6}+423792 r^{5}+1485324 r^{4}+2519317 r^{3}+2096184 r^{2}+734987 r +50388\right )}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (a_{k -5}-24 a_{k -4}+216 a_{k -3}-864 a_{k -2}+1296 a_{k -1}\right ) k^{2}+\left (2 \left (a_{k -5}-24 a_{k -4}+216 a_{k -3}-864 a_{k -2}+1296 a_{k -1}\right ) r -11 a_{k -5}+216 a_{k -4}-1511 a_{k -3}+4308 a_{k -2}-3852 a_{k -1}\right ) k +\left (a_{k -5}-24 a_{k -4}+216 a_{k -3}-864 a_{k -2}+1296 a_{k -1}\right ) r^{2}+\left (-11 a_{k -5}+216 a_{k -4}-1511 a_{k -3}+4308 a_{k -2}-3852 a_{k -1}\right ) r +30 a_{k -5}-480 a_{k -4}+2591 a_{k -3}-5184 a_{k -2}+2628 a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +5 \\ {} & {} & \left (a_{k}-24 a_{k +1}+216 a_{k +2}-864 a_{k +3}+1296 a_{k +4}\right ) \left (k +5\right )^{2}+\left (2 \left (a_{k}-24 a_{k +1}+216 a_{k +2}-864 a_{k +3}+1296 a_{k +4}\right ) r -11 a_{k}+216 a_{k +1}-1511 a_{k +2}+4308 a_{k +3}-3852 a_{k +4}\right ) \left (k +5\right )+\left (a_{k}-24 a_{k +1}+216 a_{k +2}-864 a_{k +3}+1296 a_{k +4}\right ) r^{2}+\left (-11 a_{k}+216 a_{k +1}-1511 a_{k +2}+4308 a_{k +3}-3852 a_{k +4}\right ) r +30 a_{k}-480 a_{k +1}+2591 a_{k +2}-5184 a_{k +3}+2628 a_{k +4}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +4}=-\frac {k^{2} a_{k}-24 k^{2} a_{k +1}+216 k^{2} a_{k +2}-864 k^{2} a_{k +3}+2 k r a_{k}-48 k r a_{k +1}+432 k r a_{k +2}-1728 k r a_{k +3}+r^{2} a_{k}-24 r^{2} a_{k +1}+216 r^{2} a_{k +2}-864 r^{2} a_{k +3}-k a_{k}-24 k a_{k +1}+649 k a_{k +2}-4332 k a_{k +3}-r a_{k}-24 r a_{k +1}+649 r a_{k +2}-4332 r a_{k +3}+436 a_{k +2}-5244 a_{k +3}}{36 \left (36 k^{2}+72 k r +36 r^{2}+253 k +253 r +438\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {35}{72}-\frac {\sqrt {937}}{72} \\ {} & {} & a_{k +4}=-\frac {k^{2} a_{k}-24 k^{2} a_{k +1}+216 k^{2} a_{k +2}-864 k^{2} a_{k +3}+2 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k}-48 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +1}+432 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +2}-1728 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +3}+\left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k}-24 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +1}+216 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +2}-864 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +3}-k a_{k}-24 k a_{k +1}+649 k a_{k +2}-4332 k a_{k +3}-\left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k}-24 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +1}+649 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +2}-4332 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +3}+436 a_{k +2}-5244 a_{k +3}}{36 \left (36 k^{2}+72 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )+36 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+253 k +\frac {40391}{72}-\frac {253 \sqrt {937}}{72}\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {35}{72}-\frac {\sqrt {937}}{72} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {35}{72}-\frac {\sqrt {937}}{72}}, a_{k +4}=-\frac {k^{2} a_{k}-24 k^{2} a_{k +1}+216 k^{2} a_{k +2}-864 k^{2} a_{k +3}+2 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k}-48 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +1}+432 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +2}-1728 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +3}+\left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k}-24 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +1}+216 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +2}-864 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +3}-k a_{k}-24 k a_{k +1}+649 k a_{k +2}-4332 k a_{k +3}-\left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k}-24 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +1}+649 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +2}-4332 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +3}+436 a_{k +2}-5244 a_{k +3}}{36 \left (36 k^{2}+72 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )+36 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+253 k +\frac {40391}{72}-\frac {253 \sqrt {937}}{72}\right )}, a_{1}=\frac {a_{0} \left (72 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}-\frac {2341}{72}+\frac {71 \sqrt {937}}{72}\right )}{3 \left (36 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {1511}{72}-\frac {37 \sqrt {937}}{72}\right )}, a_{2}=\frac {a_{0} \left (4320 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}+108 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}-4019 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {3967}{72}-\frac {101 \sqrt {937}}{72}\right )}{12 \left (1296 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}+5256 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}+6877 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {126281}{72}-\frac {3139 \sqrt {937}}{72}\right )}, a_{3}=\frac {a_{0} \left (233280 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{6}+707616 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{5}+277488 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}-618425 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}-414801 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {169979}{36}-\frac {3721 \sqrt {937}}{36}\right )}{54 \left (46656 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{6}+423792 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{5}+1485324 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}+2519317 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}+2096184 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {29352481}{72}-\frac {734987 \sqrt {937}}{72}\right )}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +3 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +3\right )^{k +\frac {35}{72}-\frac {\sqrt {937}}{72}}, a_{k +4}=-\frac {k^{2} a_{k}-24 k^{2} a_{k +1}+216 k^{2} a_{k +2}-864 k^{2} a_{k +3}+2 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k}-48 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +1}+432 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +2}-1728 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +3}+\left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k}-24 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +1}+216 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +2}-864 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +3}-k a_{k}-24 k a_{k +1}+649 k a_{k +2}-4332 k a_{k +3}-\left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k}-24 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +1}+649 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +2}-4332 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +3}+436 a_{k +2}-5244 a_{k +3}}{36 \left (36 k^{2}+72 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )+36 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+253 k +\frac {40391}{72}-\frac {253 \sqrt {937}}{72}\right )}, a_{1}=\frac {a_{0} \left (72 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}-\frac {2341}{72}+\frac {71 \sqrt {937}}{72}\right )}{3 \left (36 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {1511}{72}-\frac {37 \sqrt {937}}{72}\right )}, a_{2}=\frac {a_{0} \left (4320 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}+108 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}-4019 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {3967}{72}-\frac {101 \sqrt {937}}{72}\right )}{12 \left (1296 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}+5256 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}+6877 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {126281}{72}-\frac {3139 \sqrt {937}}{72}\right )}, a_{3}=\frac {a_{0} \left (233280 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{6}+707616 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{5}+277488 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}-618425 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}-414801 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {169979}{36}-\frac {3721 \sqrt {937}}{36}\right )}{54 \left (46656 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{6}+423792 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{5}+1485324 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}+2519317 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}+2096184 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {29352481}{72}-\frac {734987 \sqrt {937}}{72}\right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {35}{72}+\frac {\sqrt {937}}{72} \\ {} & {} & a_{k +4}=-\frac {k^{2} a_{k}-24 k^{2} a_{k +1}+216 k^{2} a_{k +2}-864 k^{2} a_{k +3}+2 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k}-48 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +1}+432 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +2}-1728 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +3}+\left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k}-24 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k +1}+216 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k +2}-864 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k +3}-k a_{k}-24 k a_{k +1}+649 k a_{k +2}-4332 k a_{k +3}-\left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k}-24 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +1}+649 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +2}-4332 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +3}+436 a_{k +2}-5244 a_{k +3}}{36 \left (36 k^{2}+72 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )+36 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+253 k +\frac {40391}{72}+\frac {253 \sqrt {937}}{72}\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {35}{72}+\frac {\sqrt {937}}{72} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {35}{72}+\frac {\sqrt {937}}{72}}, a_{k +4}=-\frac {k^{2} a_{k}-24 k^{2} a_{k +1}+216 k^{2} a_{k +2}-864 k^{2} a_{k +3}+2 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k}-48 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +1}+432 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +2}-1728 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +3}+\left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k}-24 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k +1}+216 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k +2}-864 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k +3}-k a_{k}-24 k a_{k +1}+649 k a_{k +2}-4332 k a_{k +3}-\left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k}-24 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +1}+649 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +2}-4332 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +3}+436 a_{k +2}-5244 a_{k +3}}{36 \left (36 k^{2}+72 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )+36 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+253 k +\frac {40391}{72}+\frac {253 \sqrt {937}}{72}\right )}, a_{1}=\frac {a_{0} \left (72 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}-\frac {2341}{72}-\frac {71 \sqrt {937}}{72}\right )}{3 \left (36 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {1511}{72}+\frac {37 \sqrt {937}}{72}\right )}, a_{2}=\frac {a_{0} \left (4320 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}+108 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}-4019 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {3967}{72}+\frac {101 \sqrt {937}}{72}\right )}{12 \left (1296 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}+5256 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}+6877 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {126281}{72}+\frac {3139 \sqrt {937}}{72}\right )}, a_{3}=\frac {a_{0} \left (233280 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{6}+707616 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{5}+277488 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}-618425 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}-414801 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {169979}{36}+\frac {3721 \sqrt {937}}{36}\right )}{54 \left (46656 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{6}+423792 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{5}+1485324 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}+2519317 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}+2096184 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {29352481}{72}+\frac {734987 \sqrt {937}}{72}\right )}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +3 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +3\right )^{k +\frac {35}{72}+\frac {\sqrt {937}}{72}}, a_{k +4}=-\frac {k^{2} a_{k}-24 k^{2} a_{k +1}+216 k^{2} a_{k +2}-864 k^{2} a_{k +3}+2 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k}-48 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +1}+432 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +2}-1728 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +3}+\left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k}-24 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k +1}+216 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k +2}-864 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} a_{k +3}-k a_{k}-24 k a_{k +1}+649 k a_{k +2}-4332 k a_{k +3}-\left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k}-24 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +1}+649 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +2}-4332 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) a_{k +3}+436 a_{k +2}-5244 a_{k +3}}{36 \left (36 k^{2}+72 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )+36 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+253 k +\frac {40391}{72}+\frac {253 \sqrt {937}}{72}\right )}, a_{1}=\frac {a_{0} \left (72 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}-\frac {2341}{72}-\frac {71 \sqrt {937}}{72}\right )}{3 \left (36 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {1511}{72}+\frac {37 \sqrt {937}}{72}\right )}, a_{2}=\frac {a_{0} \left (4320 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}+108 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}-4019 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {3967}{72}+\frac {101 \sqrt {937}}{72}\right )}{12 \left (1296 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}+5256 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}+6877 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {126281}{72}+\frac {3139 \sqrt {937}}{72}\right )}, a_{3}=\frac {a_{0} \left (233280 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{6}+707616 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{5}+277488 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}-618425 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}-414801 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {169979}{36}+\frac {3721 \sqrt {937}}{36}\right )}{54 \left (46656 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{6}+423792 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{5}+1485324 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}+2519317 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}+2096184 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {29352481}{72}+\frac {734987 \sqrt {937}}{72}\right )}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +3\right )^{k +\frac {35}{72}-\frac {\sqrt {937}}{72}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} \left (x +3\right )^{k +\frac {35}{72}+\frac {\sqrt {937}}{72}}\right ), a_{k +4}=-\frac {k^{2} a_{k}-24 k^{2} a_{k +1}+216 k^{2} a_{k +2}-864 k^{2} a_{k +3}+2 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k}-48 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +1}+432 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +2}-1728 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +3}+\left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k}-24 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +1}+216 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +2}-864 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2} a_{k +3}-k a_{k}-24 k a_{k +1}+649 k a_{k +2}-4332 k a_{k +3}-\left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k}-24 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +1}+649 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +2}-4332 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right ) a_{k +3}+436 a_{k +2}-5244 a_{k +3}}{36 \left (36 k^{2}+72 k \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )+36 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+253 k +\frac {40391}{72}-\frac {253 \sqrt {937}}{72}\right )}, a_{1}=\frac {a_{0} \left (72 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}-\frac {2341}{72}+\frac {71 \sqrt {937}}{72}\right )}{3 \left (36 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {1511}{72}-\frac {37 \sqrt {937}}{72}\right )}, a_{2}=\frac {a_{0} \left (4320 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}+108 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}-4019 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {3967}{72}-\frac {101 \sqrt {937}}{72}\right )}{12 \left (1296 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}+5256 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}+6877 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {126281}{72}-\frac {3139 \sqrt {937}}{72}\right )}, a_{3}=\frac {a_{0} \left (233280 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{6}+707616 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{5}+277488 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}-618425 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}-414801 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {169979}{36}-\frac {3721 \sqrt {937}}{36}\right )}{54 \left (46656 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{6}+423792 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{5}+1485324 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{4}+2519317 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{3}+2096184 \left (\frac {35}{72}-\frac {\sqrt {937}}{72}\right )^{2}+\frac {29352481}{72}-\frac {734987 \sqrt {937}}{72}\right )}, b_{k +4}=-\frac {k^{2} b_{k}-24 k^{2} b_{k +1}+216 k^{2} b_{k +2}-864 k^{2} b_{k +3}+2 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) b_{k}-48 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) b_{k +1}+432 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) b_{k +2}-1728 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) b_{k +3}+\left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} b_{k}-24 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} b_{k +1}+216 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} b_{k +2}-864 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2} b_{k +3}-k b_{k}-24 k b_{k +1}+649 k b_{k +2}-4332 k b_{k +3}-\left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) b_{k}-24 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) b_{k +1}+649 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) b_{k +2}-4332 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right ) b_{k +3}+436 b_{k +2}-5244 b_{k +3}}{36 \left (36 k^{2}+72 k \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )+36 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+253 k +\frac {40391}{72}+\frac {253 \sqrt {937}}{72}\right )}, b_{1}=\frac {b_{0} \left (72 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}-\frac {2341}{72}-\frac {71 \sqrt {937}}{72}\right )}{3 \left (36 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {1511}{72}+\frac {37 \sqrt {937}}{72}\right )}, b_{2}=\frac {b_{0} \left (4320 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}+108 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}-4019 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {3967}{72}+\frac {101 \sqrt {937}}{72}\right )}{12 \left (1296 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}+5256 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}+6877 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {126281}{72}+\frac {3139 \sqrt {937}}{72}\right )}, b_{3}=\frac {b_{0} \left (233280 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{6}+707616 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{5}+277488 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}-618425 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}-414801 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {169979}{36}+\frac {3721 \sqrt {937}}{36}\right )}{54 \left (46656 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{6}+423792 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{5}+1485324 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{4}+2519317 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{3}+2096184 \left (\frac {35}{72}+\frac {\sqrt {937}}{72}\right )^{2}+\frac {29352481}{72}+\frac {734987 \sqrt {937}}{72}\right )}\right ] \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Kummer successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 54
Order:=6; dsolve((x^2-9)^2*diff(y(x),x$2)+(x+3)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 70
AsymptoticDSolveValue[(x^2-9)^2*y''[x]+(x+3)*y'[x]+2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {304 x^5}{23914845}-\frac {289 x^4}{708588}+\frac {x^3}{6561}-\frac {x^2}{81}+1\right )+c_2 \left (-\frac {596 x^5}{1594323}-\frac {131 x^4}{236196}-\frac {13 x^3}{2187}-\frac {x^2}{54}+x\right ) \]