Internal problem ID [1267]
Internal file name [OUTPUT/1268_Sunday_June_05_2022_02_07_29_AM_65047568/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR
AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number: 29.
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]]
\[ \boxed {\left (\beta \,x^{2}+\alpha x +1\right ) y^{\prime \prime }+\left (\delta x +\gamma \right ) y^{\prime }+\epsilon y=0} \] 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 } \delta x +y^{\prime } \gamma +\epsilon y}{\beta \,x^{2}+\alpha x +1}\\ 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 (\left (-\beta \epsilon +\delta \left (\beta +\delta \right )\right ) x^{2}+\left (-\alpha \epsilon +2 \gamma \left (\beta +\delta \right )\right ) x +\alpha \gamma +\gamma ^{2}-\delta -\epsilon \right ) y^{\prime }+2 \left (\left (\beta +\frac {\delta }{2}\right ) x +\frac {\alpha }{2}+\frac {\gamma }{2}\right ) y \epsilon }{\left (\beta \,x^{2}+\alpha x +1\right )^{2}}\\ 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 (-2 \left (\beta +\frac {\delta }{2}\right ) \left (-2 \beta \epsilon +\delta \left (\beta +\delta \right )\right ) x^{3}+\left (\left (\left (6 \alpha +2 \gamma \right ) \beta +2 \alpha \delta \right ) \epsilon -6 \left (\beta +\frac {\delta }{2}\right ) \left (\beta +\delta \right ) \gamma \right ) x^{2}+\left (\left (2 \alpha ^{2}+2 \alpha \gamma +4 \beta +2 \delta \right ) \epsilon -6 \left (\beta +\frac {\delta }{2}\right ) \left (\alpha \gamma +\gamma ^{2}-\delta \right )\right ) x +\left (2 \alpha +2 \gamma \right ) \epsilon +2 \beta \gamma +\left (2 \alpha +3 \gamma \right ) \delta -2 \left (\alpha +\gamma \right ) \left (\alpha +\frac {\gamma }{2}\right ) \gamma \right ) y^{\prime }-6 y \epsilon \left (\left (\beta ^{2}+\frac {2}{3} \beta \delta -\frac {1}{6} \beta \epsilon +\frac {1}{6} \delta ^{2}\right ) x^{2}+\left (-\frac {\alpha \epsilon }{6}+\left (\alpha +\gamma \right ) \beta +\frac {\delta \left (\alpha +2 \gamma \right )}{6}\right ) x +\frac {\alpha ^{2}}{3}+\frac {\alpha \gamma }{2}+\frac {\gamma ^{2}}{6}-\frac {\beta }{3}-\frac {\delta }{3}-\frac {\epsilon }{6}\right )}{\left (\beta \,x^{2}+\alpha x +1\right )^{3}}\\ 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 (\left (\beta ^{2} \epsilon ^{2}+\left (-18 \beta ^{3}-14 \delta \,\beta ^{2}-3 \beta \,\delta ^{2}\right ) \epsilon +6 \beta ^{3} \delta +11 \beta ^{2} \delta ^{2}+6 \beta \,\delta ^{3}+\delta ^{4}\right ) x^{4}+\left (2 \beta \alpha \,\epsilon ^{2}+\left (\left (-36 \alpha -18 \gamma \right ) \beta ^{2}-19 \left (\alpha +\frac {6 \gamma }{19}\right ) \delta \beta -3 \alpha \,\delta ^{2}\right ) \epsilon +24 \left (\beta +\frac {\delta }{2}\right ) \left (\beta +\frac {\delta }{3}\right ) \left (\beta +\delta \right ) \gamma \right ) x^{3}+\left (\left (\alpha ^{2}+2 \beta \right ) \epsilon ^{2}+\left (-12 \beta ^{2}+\left (-24 \alpha ^{2}-27 \alpha \gamma -3 \gamma ^{2}-10 \delta \right ) \beta -3 \delta ^{2}+\left (-5 \alpha ^{2}-6 \alpha \gamma \right ) \delta \right ) \epsilon +36 \left (\beta +\frac {\delta }{2}\right ) \left (\beta +\frac {\delta }{3}\right ) \left (\alpha \gamma +\gamma ^{2}-\delta \right )\right ) x^{2}+\left (2 \alpha \,\epsilon ^{2}+\left (\left (-12 \alpha -18 \gamma \right ) \beta +\left (-\alpha -6 \gamma \right ) \delta -6 \alpha \left (\alpha +\gamma \right ) \left (\alpha +\frac {\gamma }{2}\right )\right ) \epsilon -24 \left (\beta \gamma +\left (\alpha +\frac {3 \gamma }{2}\right ) \delta -\left (\alpha +\gamma \right ) \left (\alpha +\frac {\gamma }{2}\right ) \gamma \right ) \left (\beta +\frac {\delta }{3}\right )\right ) x +\epsilon ^{2}+\left (-6 \alpha ^{2}-9 \alpha \gamma -3 \gamma ^{2}+6 \beta +4 \delta \right ) \epsilon +\left (-12 \alpha \gamma -8 \gamma ^{2}+6 \delta \right ) \beta +3 \delta ^{2}+\left (-6 \alpha ^{2}-14 \alpha \gamma -6 \gamma ^{2}\right ) \delta +6 \alpha ^{3} \gamma +11 \alpha ^{2} \gamma ^{2}+6 \alpha \,\gamma ^{3}+\gamma ^{4}\right ) y^{\prime }+24 \left (\left (\beta +\frac {\delta }{4}\right ) \left (\beta ^{2}+\frac {1}{2} \beta \delta -\frac {1}{3} \beta \epsilon +\frac {1}{6} \delta ^{2}\right ) x^{3}+\left (\left (\left (-\frac {\alpha }{2}-\frac {\gamma }{12}\right ) \beta -\frac {\alpha \delta }{12}\right ) \epsilon +\left (\frac {3 \gamma }{2}+\frac {3 \alpha }{2}\right ) \beta ^{2}+\frac {3 \delta \left (\frac {19 \gamma }{9}+\alpha \right ) \beta }{8}+\frac {\delta ^{2} \left (3 \gamma +\alpha \right )}{24}\right ) x^{2}+\left (\left (-\frac {1}{12} \delta -\frac {1}{6} \alpha ^{2}-\frac {1}{3} \beta -\frac {1}{12} \alpha \gamma \right ) \epsilon -\beta ^{2}+\left (\alpha ^{2}+\frac {1}{2} \gamma ^{2}-\frac {13}{12} \delta +\frac {3}{2} \alpha \gamma \right ) \beta +\frac {\delta \left (-\frac {5 \delta }{2}+\left (3 \gamma +\alpha \right ) \left (\alpha +\frac {\gamma }{2}\right )\right )}{12}\right ) x +\left (-\frac {\gamma }{12}-\frac {\alpha }{6}\right ) \epsilon +\left (-\frac {\gamma }{3}-\frac {\alpha }{2}\right ) \beta +\left (-\frac {3 \alpha }{8}-\frac {5 \gamma }{24}\right ) \delta +\frac {\left (\alpha +\gamma \right ) \left (\alpha +\frac {\gamma }{2}\right ) \left (\alpha +\frac {\gamma }{3}\right )}{4}\right ) y \epsilon }{\left (\beta \,x^{2}+\alpha x +1\right )^{4}}\\ 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 \\ &= \text {Expression too large to display} \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 &= -y \left (0\right ) \epsilon -y^{\prime }\left (0\right ) \gamma \\ F_1 &= y \left (0\right ) \alpha \epsilon +y \left (0\right ) \epsilon \gamma +y^{\prime }\left (0\right ) \alpha \gamma +y^{\prime }\left (0\right ) \gamma ^{2}-y^{\prime }\left (0\right ) \delta -y^{\prime }\left (0\right ) \epsilon \\ F_2 &= -2 y \left (0\right ) \alpha ^{2} \epsilon -3 y \left (0\right ) \alpha \epsilon \gamma -y \left (0\right ) \epsilon \,\gamma ^{2}-2 y^{\prime }\left (0\right ) \alpha ^{2} \gamma -3 y^{\prime }\left (0\right ) \alpha \,\gamma ^{2}-y^{\prime }\left (0\right ) \gamma ^{3}+2 y \left (0\right ) \beta \epsilon +2 y \left (0\right ) \epsilon \delta +\epsilon ^{2} y \left (0\right )+2 y^{\prime }\left (0\right ) \alpha \delta +2 y^{\prime }\left (0\right ) \alpha \epsilon +2 y^{\prime }\left (0\right ) \beta \gamma +3 y^{\prime }\left (0\right ) \delta \gamma +2 y^{\prime }\left (0\right ) \epsilon \gamma \\ F_3 &= -6 y^{\prime }\left (0\right ) \delta \,\gamma ^{2}-3 y^{\prime }\left (0\right ) \epsilon \,\gamma ^{2}+6 y^{\prime }\left (0\right ) \beta \epsilon +4 y^{\prime }\left (0\right ) \delta \epsilon +11 y^{\prime }\left (0\right ) \alpha ^{2} \gamma ^{2}+3 y^{\prime }\left (0\right ) \delta ^{2}+y^{\prime }\left (0\right ) \gamma ^{4}+y^{\prime }\left (0\right ) \epsilon ^{2}+6 y^{\prime }\left (0\right ) \alpha ^{3} \gamma -6 y^{\prime }\left (0\right ) \alpha ^{2} \delta +6 y^{\prime }\left (0\right ) \beta \delta +11 y \left (0\right ) \alpha ^{2} \epsilon \gamma +6 y \left (0\right ) \alpha \epsilon \,\gamma ^{2}-9 y \left (0\right ) \alpha \epsilon \delta +6 y^{\prime }\left (0\right ) \alpha \,\gamma ^{3}-6 y^{\prime }\left (0\right ) \alpha ^{2} \epsilon -8 y^{\prime }\left (0\right ) \beta \,\gamma ^{2}-12 y \left (0\right ) \alpha \beta \epsilon -12 y^{\prime }\left (0\right ) \alpha \beta \gamma +6 y \left (0\right ) \alpha ^{3} \epsilon -8 y \left (0\right ) \beta \epsilon \gamma -5 y \left (0\right ) \epsilon \delta \gamma +y \left (0\right ) \epsilon \,\gamma ^{3}-4 \epsilon ^{2} y \left (0\right ) \alpha -2 \epsilon ^{2} y \left (0\right ) \gamma -14 y^{\prime }\left (0\right ) \alpha \delta \gamma -9 y^{\prime }\left (0\right ) \alpha \epsilon \gamma \\ F_4 &= -y^{\prime }\left (0\right ) \gamma ^{5}-32 y^{\prime }\left (0\right ) \beta \epsilon \gamma -15 y^{\prime }\left (0\right ) \delta \epsilon \gamma +70 y^{\prime }\left (0\right ) \alpha ^{2} \delta \gamma +44 y^{\prime }\left (0\right ) \alpha ^{2} \epsilon \gamma +80 y^{\prime }\left (0\right ) \alpha \beta \,\gamma ^{2}+50 y^{\prime }\left (0\right ) \alpha \delta \,\gamma ^{2}+24 y^{\prime }\left (0\right ) \alpha \epsilon \,\gamma ^{2}-24 y \left (0\right ) \alpha ^{4} \epsilon -24 y^{\prime }\left (0\right ) \alpha ^{4} \gamma +24 y^{\prime }\left (0\right ) \alpha ^{3} \delta -24 y \left (0\right ) \beta ^{2} \epsilon -24 y^{\prime }\left (0\right ) \beta ^{2} \gamma -y \left (0\right ) \epsilon \,\gamma ^{4}+18 \epsilon ^{2} y \left (0\right ) \alpha ^{2}+3 \epsilon ^{2} y \left (0\right ) \gamma ^{2}-14 \epsilon ^{2} y \left (0\right ) \beta -8 y \left (0\right ) \epsilon \,\delta ^{2}-6 \epsilon ^{2} y \left (0\right ) \delta -6 y^{\prime }\left (0\right ) \alpha \,\epsilon ^{2}-15 y^{\prime }\left (0\right ) \delta ^{2} \gamma -3 y^{\prime }\left (0\right ) \epsilon ^{2} \gamma -50 y^{\prime }\left (0\right ) \alpha ^{3} \gamma ^{2}-35 y^{\prime }\left (0\right ) \alpha ^{2} \gamma ^{3}-10 y^{\prime }\left (0\right ) \alpha \,\gamma ^{4}+24 y^{\prime }\left (0\right ) \alpha ^{3} \epsilon -\epsilon ^{3} y \left (0\right )+20 y^{\prime }\left (0\right ) \beta \,\gamma ^{3}+10 y^{\prime }\left (0\right ) \delta \,\gamma ^{3}+4 y^{\prime }\left (0\right ) \epsilon \,\gamma ^{3}-20 y^{\prime }\left (0\right ) \alpha \,\delta ^{2}-48 y^{\prime }\left (0\right ) \alpha \beta \epsilon -26 y^{\prime }\left (0\right ) \alpha \delta \epsilon -50 y^{\prime }\left (0\right ) \beta \delta \gamma +72 y \left (0\right ) \alpha ^{2} \beta \epsilon +72 y^{\prime }\left (0\right ) \alpha ^{2} \beta \gamma -48 y^{\prime }\left (0\right ) \alpha \beta \delta +80 y \left (0\right ) \alpha \beta \epsilon \gamma +41 y \left (0\right ) \alpha \epsilon \delta \gamma -50 y \left (0\right ) \alpha ^{3} \epsilon \gamma -35 y \left (0\right ) \alpha ^{2} \epsilon \,\gamma ^{2}-10 y \left (0\right ) \alpha \epsilon \,\gamma ^{3}+44 y \left (0\right ) \alpha ^{2} \epsilon \delta +15 \epsilon ^{2} y \left (0\right ) \gamma \alpha +20 y \left (0\right ) \beta \epsilon \,\gamma ^{2}+9 y \left (0\right ) \epsilon \delta \,\gamma ^{2}-32 y \left (0\right ) \beta \epsilon \delta \end {align*}
Substituting all the above in (7) and simplifying gives the solution as \[ \text {Expression too large to display} \] Since the expansion point \(x = 0\) is an ordinary, we can also solve this using standard power series The ode is normalized to be \[ \left (\beta \,x^{2}+\alpha x +1\right ) y^{\prime \prime }+\left (\delta x +\gamma \right ) y^{\prime }+\epsilon y = 0 \] 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 (\beta \,x^{2}+\alpha x +1\right ) \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} x^{n -2}\right )+\left (\delta x +\gamma \right ) \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\right )+\epsilon \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 }}x^{n} a_{n} \beta n \left (n -1\right )\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}n \alpha \,x^{n -1} a_{n} \left (n -1\right )\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} x^{n -2}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n} \delta \right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}n \gamma \,x^{n -1} a_{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\epsilon 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 \alpha \,x^{n -1} a_{n} \left (n -1\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\alpha \left (n +1\right ) a_{n +1} n \,x^{n} \\ \moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} x^{n -2} &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +2\right ) a_{n +2} \left (n +1\right ) x^{n} \\ \moverset {\infty }{\munderset {n =1}{\sum }}n \gamma \,x^{n -1} a_{n} &= \moverset {\infty }{\munderset {n =0}{\sum }}\gamma \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 =2}{\sum }}x^{n} a_{n} \beta n \left (n -1\right )\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}\alpha \left (n +1\right ) a_{n +1} n \,x^{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\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} \delta \right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\gamma \left (n +1\right ) a_{n +1} x^{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\epsilon a_{n} x^{n}\right ) = 0 \end{equation} \(n=0\) gives \[ \epsilon a_{0}+\gamma a_{1}+2 a_{2}=0 \] \[ a_{2} = -\frac {\epsilon a_{0}}{2}-\frac {\gamma a_{1}}{2} \] \(n=1\) gives \[ 2 \alpha a_{2}+\delta a_{1}+\epsilon a_{1}+2 \gamma a_{2}+6 a_{3}=0 \] Which after substituting earlier equations, simplifies to \[ a_{3} = \frac {1}{6} \alpha \epsilon a_{0}+\frac {1}{6} \alpha \gamma a_{1}+\frac {1}{6} \gamma \epsilon a_{0}+\frac {1}{6} \gamma ^{2} a_{1}-\frac {1}{6} \delta a_{1}-\frac {1}{6} \epsilon a_{1} \] For \(2\le n\), the recurrence equation is \begin{equation} \tag{4} \beta n a_{n} \left (n -1\right )+\alpha \left (n +1\right ) a_{n +1} n +\left (n +2\right ) a_{n +2} \left (n +1\right )+\delta n a_{n}+\gamma \left (n +1\right ) a_{n +1}+\epsilon a_{n} = 0 \end{equation} Solving for \(a_{n +2}\), gives \begin{align*} \tag{5} a_{n +2}&= -\frac {\alpha \,n^{2} a_{n +1}+\beta \,n^{2} a_{n}+\alpha n a_{n +1}-\beta n a_{n}+\delta n a_{n}+\gamma n a_{n +1}+\epsilon a_{n}+\gamma a_{n +1}}{\left (n +2\right ) \left (n +1\right )} \\ &= -\frac {\left (\beta \,n^{2}-\beta n +\delta n +\epsilon \right ) a_{n}}{\left (n +2\right ) \left (n +1\right )}-\frac {\left (\alpha \,n^{2}+\alpha n +\gamma n +\gamma \right ) a_{n +1}}{\left (n +2\right ) \left (n +1\right )} \\ \end{align*} For \(n = 2\) the recurrence equation gives \[ 6 \alpha a_{3}+2 \beta a_{2}+2 \delta a_{2}+\epsilon a_{2}+3 \gamma a_{3}+12 a_{4} = 0 \] Which after substituting the earlier terms found becomes \[ a_{4} = -\frac {1}{12} \alpha ^{2} \epsilon a_{0}-\frac {1}{12} \alpha ^{2} \gamma a_{1}-\frac {1}{8} \alpha \gamma \epsilon a_{0}-\frac {1}{8} \alpha \,\gamma ^{2} a_{1}+\frac {1}{12} \alpha \delta a_{1}+\frac {1}{12} \alpha \epsilon a_{1}+\frac {1}{12} \beta \epsilon a_{0}+\frac {1}{12} \beta \gamma a_{1}+\frac {1}{12} \delta \epsilon a_{0}+\frac {1}{8} \delta \gamma a_{1}+\frac {1}{24} \epsilon ^{2} a_{0}+\frac {1}{12} \epsilon \gamma a_{1}-\frac {1}{24} \gamma ^{2} \epsilon a_{0}-\frac {1}{24} \gamma ^{3} a_{1} \] For \(n = 3\) the recurrence equation gives \[ 12 \alpha a_{4}+6 \beta a_{3}+3 \delta a_{3}+\epsilon a_{3}+4 \gamma a_{4}+20 a_{5} = 0 \] Which after substituting the earlier terms found becomes \[ a_{5} = \frac {1}{20} \beta \epsilon a_{1}-\frac {1}{20} \delta \,\gamma ^{2} a_{1}+\frac {1}{30} \delta \epsilon a_{1}-\frac {1}{60} \gamma \,\epsilon ^{2} a_{0}-\frac {1}{40} \epsilon \,\gamma ^{2} a_{1}+\frac {1}{120} \gamma ^{3} \epsilon a_{0}+\frac {1}{20} \alpha ^{3} \epsilon a_{0}+\frac {1}{20} \alpha ^{3} \gamma a_{1}+\frac {11}{120} \alpha ^{2} \gamma ^{2} a_{1}-\frac {1}{20} \alpha ^{2} \delta a_{1}-\frac {1}{20} \alpha ^{2} \epsilon a_{1}-\frac {1}{30} \alpha \,\epsilon ^{2} a_{0}+\frac {1}{20} \alpha \,\gamma ^{3} a_{1}-\frac {1}{15} \beta \,\gamma ^{2} a_{1}+\frac {1}{20} \beta \delta a_{1}+\frac {11}{120} \alpha ^{2} \gamma \epsilon a_{0}-\frac {1}{10} \alpha \beta \epsilon a_{0}-\frac {1}{10} \alpha \beta \gamma a_{1}-\frac {3}{40} \alpha \delta \epsilon a_{0}-\frac {7}{60} \alpha \delta \gamma a_{1}-\frac {3}{40} \alpha \epsilon \gamma a_{1}+\frac {1}{20} \alpha \,\gamma ^{2} \epsilon a_{0}-\frac {1}{15} \beta \gamma \epsilon a_{0}-\frac {1}{24} \delta \gamma \epsilon a_{0}+\frac {1}{40} \delta ^{2} a_{1}+\frac {1}{120} \epsilon ^{2} a_{1}+\frac {1}{120} \gamma ^{4} a_{1} \] For \(n = 4\) the recurrence equation gives \[ 20 \alpha a_{5}+12 \beta a_{4}+4 \delta a_{4}+\epsilon a_{4}+5 \gamma a_{5}+30 a_{6} = 0 \] Which after substituting the earlier terms found becomes \[ a_{6} = -\frac {1}{30} \alpha ^{4} \epsilon a_{0}-\frac {1}{30} \alpha ^{4} \gamma a_{1}-\frac {5}{72} \alpha ^{3} \gamma ^{2} a_{1}+\frac {1}{30} \alpha ^{3} \delta a_{1}+\frac {1}{30} \alpha ^{3} \epsilon a_{1}+\frac {1}{40} \alpha ^{2} \epsilon ^{2} a_{0}-\frac {7}{144} \alpha ^{2} \gamma ^{3} a_{1}-\frac {1}{36} \alpha \,\delta ^{2} a_{1}-\frac {1}{120} \alpha \,\epsilon ^{2} a_{1}-\frac {1}{72} \alpha \,\gamma ^{3} \epsilon a_{0}+\frac {1}{9} \alpha \beta \,\gamma ^{2} a_{1}-\frac {1}{15} \alpha \beta \delta a_{1}-\frac {5}{72} \alpha ^{3} \gamma \epsilon a_{0}+\frac {1}{10} \alpha ^{2} \beta \epsilon a_{0}+\frac {1}{10} \alpha ^{2} \beta \gamma a_{1}+\frac {11}{180} \alpha ^{2} \delta \epsilon a_{0}+\frac {7}{72} \alpha ^{2} \delta \gamma a_{1}+\frac {11}{180} \alpha ^{2} \epsilon \gamma a_{1}-\frac {7}{144} \alpha ^{2} \gamma ^{2} \epsilon a_{0}-\frac {2}{45} \beta \delta \epsilon a_{0}-\frac {5}{72} \beta \delta \gamma a_{1}-\frac {2}{45} \beta \epsilon \gamma a_{1}+\frac {1}{36} \beta \,\gamma ^{2} \epsilon a_{0}-\frac {1}{48} \delta \epsilon \gamma a_{1}+\frac {1}{80} \delta \,\gamma ^{2} \epsilon a_{0}-\frac {1}{72} \alpha \,\gamma ^{4} a_{1}-\frac {1}{30} \beta ^{2} \epsilon a_{0}-\frac {1}{30} \beta ^{2} \gamma a_{1}-\frac {7}{360} \beta \,\epsilon ^{2} a_{0}+\frac {1}{36} \beta \,\gamma ^{3} a_{1}-\frac {1}{90} \delta ^{2} \epsilon a_{0}-\frac {1}{48} \delta ^{2} \gamma a_{1}-\frac {1}{120} \delta \,\epsilon ^{2} a_{0}+\frac {1}{72} \delta \,\gamma ^{3} a_{1}-\frac {1}{240} \epsilon ^{2} \gamma a_{1}+\frac {1}{240} \gamma ^{2} \epsilon ^{2} a_{0}+\frac {1}{180} \epsilon \,\gamma ^{3} a_{1}-\frac {1}{720} \gamma ^{4} \epsilon a_{0}-\frac {1}{15} \alpha \beta \epsilon a_{1}+\frac {5}{72} \alpha \delta \,\gamma ^{2} a_{1}-\frac {13}{360} \alpha \delta \epsilon a_{1}+\frac {1}{48} \alpha \gamma \,\epsilon ^{2} a_{0}+\frac {1}{30} \alpha \epsilon \,\gamma ^{2} a_{1}-\frac {1}{720} \epsilon ^{3} a_{0}-\frac {1}{720} \gamma ^{5} a_{1}+\frac {1}{9} \alpha \beta \gamma \epsilon a_{0}+\frac {41}{720} \alpha \delta \gamma \epsilon a_{0} \] For \(n = 5\) the recurrence equation gives \[ 30 \alpha a_{6}+20 \beta a_{5}+5 \delta a_{5}+\epsilon a_{5}+6 \gamma a_{6}+42 a_{7} = 0 \] Which after substituting the earlier terms found becomes \[ \text {Expression too large to display} \] 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 \[ y = a_{0}+a_{1} x +\left (-\frac {\epsilon a_{0}}{2}-\frac {\gamma a_{1}}{2}\right ) x^{2}+\left (\frac {1}{6} \alpha \epsilon a_{0}+\frac {1}{6} \alpha \gamma a_{1}+\frac {1}{6} \gamma \epsilon a_{0}+\frac {1}{6} \gamma ^{2} a_{1}-\frac {1}{6} \delta a_{1}-\frac {1}{6} \epsilon a_{1}\right ) x^{3}+\left (-\frac {1}{12} \alpha ^{2} \epsilon a_{0}-\frac {1}{12} \alpha ^{2} \gamma a_{1}-\frac {1}{8} \alpha \gamma \epsilon a_{0}-\frac {1}{8} \alpha \,\gamma ^{2} a_{1}+\frac {1}{12} \alpha \delta a_{1}+\frac {1}{12} \alpha \epsilon a_{1}+\frac {1}{12} \beta \epsilon a_{0}+\frac {1}{12} \beta \gamma a_{1}+\frac {1}{12} \delta \epsilon a_{0}+\frac {1}{8} \delta \gamma a_{1}+\frac {1}{24} \epsilon ^{2} a_{0}+\frac {1}{12} \epsilon \gamma a_{1}-\frac {1}{24} \gamma ^{2} \epsilon a_{0}-\frac {1}{24} \gamma ^{3} a_{1}\right ) x^{4}+\left (\frac {1}{20} \beta \epsilon a_{1}-\frac {1}{20} \delta \,\gamma ^{2} a_{1}+\frac {1}{30} \delta \epsilon a_{1}-\frac {1}{60} \gamma \,\epsilon ^{2} a_{0}-\frac {1}{40} \epsilon \,\gamma ^{2} a_{1}+\frac {1}{120} \gamma ^{3} \epsilon a_{0}+\frac {1}{20} \alpha ^{3} \epsilon a_{0}+\frac {1}{20} \alpha ^{3} \gamma a_{1}+\frac {11}{120} \alpha ^{2} \gamma ^{2} a_{1}-\frac {1}{20} \alpha ^{2} \delta a_{1}-\frac {1}{20} \alpha ^{2} \epsilon a_{1}-\frac {1}{30} \alpha \,\epsilon ^{2} a_{0}+\frac {1}{20} \alpha \,\gamma ^{3} a_{1}-\frac {1}{15} \beta \,\gamma ^{2} a_{1}+\frac {1}{20} \beta \delta a_{1}+\frac {11}{120} \alpha ^{2} \gamma \epsilon a_{0}-\frac {1}{10} \alpha \beta \epsilon a_{0}-\frac {1}{10} \alpha \beta \gamma a_{1}-\frac {3}{40} \alpha \delta \epsilon a_{0}-\frac {7}{60} \alpha \delta \gamma a_{1}-\frac {3}{40} \alpha \epsilon \gamma a_{1}+\frac {1}{20} \alpha \,\gamma ^{2} \epsilon a_{0}-\frac {1}{15} \beta \gamma \epsilon a_{0}-\frac {1}{24} \delta \gamma \epsilon a_{0}+\frac {1}{40} \delta ^{2} a_{1}+\frac {1}{120} \epsilon ^{2} a_{1}+\frac {1}{120} \gamma ^{4} a_{1}\right ) x^{5}+\dots \] Collecting terms, the solution becomes \begin{equation} \tag{3} y = \left (1-\frac {\epsilon \,x^{2}}{2}+\left (\frac {1}{6} \alpha \epsilon +\frac {1}{6} \epsilon \gamma \right ) x^{3}+\left (-\frac {1}{12} \epsilon \,\alpha ^{2}-\frac {1}{8} \epsilon \gamma \alpha +\frac {1}{12} \beta \epsilon +\frac {1}{12} \delta \epsilon +\frac {1}{24} \epsilon ^{2}-\frac {1}{24} \epsilon \,\gamma ^{2}\right ) x^{4}+\left (-\frac {1}{60} \epsilon ^{2} \gamma +\frac {1}{120} \epsilon \,\gamma ^{3}+\frac {1}{20} \epsilon \,\alpha ^{3}-\frac {1}{30} \alpha \,\epsilon ^{2}+\frac {11}{120} \epsilon \gamma \,\alpha ^{2}-\frac {1}{10} \beta \alpha \epsilon -\frac {3}{40} \alpha \delta \epsilon +\frac {1}{20} \epsilon \,\gamma ^{2} \alpha -\frac {1}{15} \beta \epsilon \gamma -\frac {1}{24} \delta \epsilon \gamma \right ) x^{5}\right ) a_{0}+\left (x -\frac {x^{2} \gamma }{2}+\left (\frac {1}{6} \alpha \gamma +\frac {1}{6} \gamma ^{2}-\frac {1}{6} \delta -\frac {1}{6} \epsilon \right ) x^{3}+\left (-\frac {1}{12} \alpha ^{2} \gamma -\frac {1}{8} \alpha \,\gamma ^{2}+\frac {1}{12} \alpha \delta +\frac {1}{12} \alpha \epsilon +\frac {1}{12} \beta \gamma +\frac {1}{8} \delta \gamma +\frac {1}{12} \epsilon \gamma -\frac {1}{24} \gamma ^{3}\right ) x^{4}+\left (\frac {1}{20} \beta \epsilon -\frac {1}{20} \delta \,\gamma ^{2}+\frac {1}{30} \delta \epsilon -\frac {1}{40} \epsilon \,\gamma ^{2}+\frac {1}{20} \alpha ^{3} \gamma +\frac {11}{120} \alpha ^{2} \gamma ^{2}-\frac {1}{20} \alpha ^{2} \delta -\frac {1}{20} \epsilon \,\alpha ^{2}+\frac {1}{20} \alpha \,\gamma ^{3}-\frac {1}{15} \beta \,\gamma ^{2}+\frac {1}{20} \beta \delta -\frac {1}{10} \alpha \beta \gamma -\frac {7}{60} \alpha \delta \gamma -\frac {3}{40} \epsilon \gamma \alpha +\frac {1}{40} \delta ^{2}+\frac {1}{120} \epsilon ^{2}+\frac {1}{120} \gamma ^{4}\right ) x^{5}\right ) a_{1}+O\left (x^{6}\right ) \end{equation} At \(x = 0\) the solution above becomes \[ y = \left (1-\frac {\epsilon \,x^{2}}{2}+\left (\frac {1}{6} \alpha \epsilon +\frac {1}{6} \epsilon \gamma \right ) x^{3}+\left (-\frac {1}{12} \epsilon \,\alpha ^{2}-\frac {1}{8} \epsilon \gamma \alpha +\frac {1}{12} \beta \epsilon +\frac {1}{12} \delta \epsilon +\frac {1}{24} \epsilon ^{2}-\frac {1}{24} \epsilon \,\gamma ^{2}\right ) x^{4}+\left (-\frac {1}{60} \epsilon ^{2} \gamma +\frac {1}{120} \epsilon \,\gamma ^{3}+\frac {1}{20} \epsilon \,\alpha ^{3}-\frac {1}{30} \alpha \,\epsilon ^{2}+\frac {11}{120} \epsilon \gamma \,\alpha ^{2}-\frac {1}{10} \beta \alpha \epsilon -\frac {3}{40} \alpha \delta \epsilon +\frac {1}{20} \epsilon \,\gamma ^{2} \alpha -\frac {1}{15} \beta \epsilon \gamma -\frac {1}{24} \delta \epsilon \gamma \right ) x^{5}\right ) c_{1} +\left (x -\frac {x^{2} \gamma }{2}+\left (\frac {1}{6} \alpha \gamma +\frac {1}{6} \gamma ^{2}-\frac {1}{6} \delta -\frac {1}{6} \epsilon \right ) x^{3}+\left (-\frac {1}{12} \alpha ^{2} \gamma -\frac {1}{8} \alpha \,\gamma ^{2}+\frac {1}{12} \alpha \delta +\frac {1}{12} \alpha \epsilon +\frac {1}{12} \beta \gamma +\frac {1}{8} \delta \gamma +\frac {1}{12} \epsilon \gamma -\frac {1}{24} \gamma ^{3}\right ) x^{4}+\left (\frac {1}{20} \beta \epsilon -\frac {1}{20} \delta \,\gamma ^{2}+\frac {1}{30} \delta \epsilon -\frac {1}{40} \epsilon \,\gamma ^{2}+\frac {1}{20} \alpha ^{3} \gamma +\frac {11}{120} \alpha ^{2} \gamma ^{2}-\frac {1}{20} \alpha ^{2} \delta -\frac {1}{20} \epsilon \,\alpha ^{2}+\frac {1}{20} \alpha \,\gamma ^{3}-\frac {1}{15} \beta \,\gamma ^{2}+\frac {1}{20} \beta \delta -\frac {1}{10} \alpha \beta \gamma -\frac {7}{60} \alpha \delta \gamma -\frac {3}{40} \epsilon \gamma \alpha +\frac {1}{40} \delta ^{2}+\frac {1}{120} \epsilon ^{2}+\frac {1}{120} \gamma ^{4}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \]
The solution(s) found are the following \begin{align*} \tag{1} \text {Expression too large to display} \\ \tag{2} y &= \left (1-\frac {\epsilon \,x^{2}}{2}+\left (\frac {1}{6} \alpha \epsilon +\frac {1}{6} \epsilon \gamma \right ) x^{3}+\left (-\frac {1}{12} \epsilon \,\alpha ^{2}-\frac {1}{8} \epsilon \gamma \alpha +\frac {1}{12} \beta \epsilon +\frac {1}{12} \delta \epsilon +\frac {1}{24} \epsilon ^{2}-\frac {1}{24} \epsilon \,\gamma ^{2}\right ) x^{4}+\left (-\frac {1}{60} \epsilon ^{2} \gamma +\frac {1}{120} \epsilon \,\gamma ^{3}+\frac {1}{20} \epsilon \,\alpha ^{3}-\frac {1}{30} \alpha \,\epsilon ^{2}+\frac {11}{120} \epsilon \gamma \,\alpha ^{2}-\frac {1}{10} \beta \alpha \epsilon -\frac {3}{40} \alpha \delta \epsilon +\frac {1}{20} \epsilon \,\gamma ^{2} \alpha -\frac {1}{15} \beta \epsilon \gamma -\frac {1}{24} \delta \epsilon \gamma \right ) x^{5}\right ) c_{1} +\left (x -\frac {x^{2} \gamma }{2}+\left (\frac {1}{6} \alpha \gamma +\frac {1}{6} \gamma ^{2}-\frac {1}{6} \delta -\frac {1}{6} \epsilon \right ) x^{3}+\left (-\frac {1}{12} \alpha ^{2} \gamma -\frac {1}{8} \alpha \,\gamma ^{2}+\frac {1}{12} \alpha \delta +\frac {1}{12} \alpha \epsilon +\frac {1}{12} \beta \gamma +\frac {1}{8} \delta \gamma +\frac {1}{12} \epsilon \gamma -\frac {1}{24} \gamma ^{3}\right ) x^{4}+\left (\frac {1}{20} \beta \epsilon -\frac {1}{20} \delta \,\gamma ^{2}+\frac {1}{30} \delta \epsilon -\frac {1}{40} \epsilon \,\gamma ^{2}+\frac {1}{20} \alpha ^{3} \gamma +\frac {11}{120} \alpha ^{2} \gamma ^{2}-\frac {1}{20} \alpha ^{2} \delta -\frac {1}{20} \epsilon \,\alpha ^{2}+\frac {1}{20} \alpha \,\gamma ^{3}-\frac {1}{15} \beta \,\gamma ^{2}+\frac {1}{20} \beta \delta -\frac {1}{10} \alpha \beta \gamma -\frac {7}{60} \alpha \delta \gamma -\frac {3}{40} \epsilon \gamma \alpha +\frac {1}{40} \delta ^{2}+\frac {1}{120} \epsilon ^{2}+\frac {1}{120} \gamma ^{4}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \\ \end{align*}
Verification of solutions
\[ \text {Expression too large to display} \] Verified OK.
\[ y = \left (1-\frac {\epsilon \,x^{2}}{2}+\left (\frac {1}{6} \alpha \epsilon +\frac {1}{6} \epsilon \gamma \right ) x^{3}+\left (-\frac {1}{12} \epsilon \,\alpha ^{2}-\frac {1}{8} \epsilon \gamma \alpha +\frac {1}{12} \beta \epsilon +\frac {1}{12} \delta \epsilon +\frac {1}{24} \epsilon ^{2}-\frac {1}{24} \epsilon \,\gamma ^{2}\right ) x^{4}+\left (-\frac {1}{60} \epsilon ^{2} \gamma +\frac {1}{120} \epsilon \,\gamma ^{3}+\frac {1}{20} \epsilon \,\alpha ^{3}-\frac {1}{30} \alpha \,\epsilon ^{2}+\frac {11}{120} \epsilon \gamma \,\alpha ^{2}-\frac {1}{10} \beta \alpha \epsilon -\frac {3}{40} \alpha \delta \epsilon +\frac {1}{20} \epsilon \,\gamma ^{2} \alpha -\frac {1}{15} \beta \epsilon \gamma -\frac {1}{24} \delta \epsilon \gamma \right ) x^{5}\right ) c_{1} +\left (x -\frac {x^{2} \gamma }{2}+\left (\frac {1}{6} \alpha \gamma +\frac {1}{6} \gamma ^{2}-\frac {1}{6} \delta -\frac {1}{6} \epsilon \right ) x^{3}+\left (-\frac {1}{12} \alpha ^{2} \gamma -\frac {1}{8} \alpha \,\gamma ^{2}+\frac {1}{12} \alpha \delta +\frac {1}{12} \alpha \epsilon +\frac {1}{12} \beta \gamma +\frac {1}{8} \delta \gamma +\frac {1}{12} \epsilon \gamma -\frac {1}{24} \gamma ^{3}\right ) x^{4}+\left (\frac {1}{20} \beta \epsilon -\frac {1}{20} \delta \,\gamma ^{2}+\frac {1}{30} \delta \epsilon -\frac {1}{40} \epsilon \,\gamma ^{2}+\frac {1}{20} \alpha ^{3} \gamma +\frac {11}{120} \alpha ^{2} \gamma ^{2}-\frac {1}{20} \alpha ^{2} \delta -\frac {1}{20} \epsilon \,\alpha ^{2}+\frac {1}{20} \alpha \,\gamma ^{3}-\frac {1}{15} \beta \,\gamma ^{2}+\frac {1}{20} \beta \delta -\frac {1}{10} \alpha \beta \gamma -\frac {7}{60} \alpha \delta \gamma -\frac {3}{40} \epsilon \gamma \alpha +\frac {1}{40} \delta ^{2}+\frac {1}{120} \epsilon ^{2}+\frac {1}{120} \gamma ^{4}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\beta \,x^{2}+\alpha x +1\right ) y^{\prime \prime }+\left (\delta x +\gamma \right ) y^{\prime }+\epsilon 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 {\epsilon y}{\beta \,x^{2}+\alpha x +1}-\frac {\left (\delta x +\gamma \right ) y^{\prime }}{\beta \,x^{2}+\alpha x +1} \\ \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 {\left (\delta x +\gamma \right ) y^{\prime }}{\beta \,x^{2}+\alpha x +1}+\frac {\epsilon y}{\beta \,x^{2}+\alpha x +1}=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 {\delta x +\gamma }{\beta \,x^{2}+\alpha x +1}, P_{3}\left (x \right )=\frac {\epsilon }{\beta \,x^{2}+\alpha x +1}\right ] \\ {} & \circ & \left (x -\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta } \\ {} & {} & \left (\left (x -\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }}}}=0 \\ {} & \circ & {\left (x -\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }\right )}^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta } \\ {} & {} & \left ({\left (x -\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }\right )}^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }}}}=0 \\ {} & \circ & x =\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }\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}=\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta } \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (\beta \,x^{2}+\alpha x +1\right ) y^{\prime \prime }+\left (\delta x +\gamma \right ) y^{\prime }+\epsilon y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u +\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (\beta \,u^{2}+u \sqrt {\alpha ^{2}-4 \beta }\right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (\delta u -\frac {\delta \alpha }{2 \beta }+\frac {\delta \sqrt {\alpha ^{2}-4 \beta }}{2 \beta }+\gamma \right ) \left (\frac {d}{d u}y \left (u \right )\right )+\epsilon 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 \left (\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & 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 =1..2 \\ {} & {} & 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} \\ {} & {} & \frac {a_{0} r \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta r -2 \sqrt {\alpha ^{2}-4 \beta }\, \beta +\sqrt {\alpha ^{2}-4 \beta }\, \delta -\alpha \delta +2 \beta \gamma \right ) u^{-1+r}}{2 \beta }+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (\frac {a_{k +1} \left (k +1+r \right ) \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \left (k +1\right )+2 \sqrt {\alpha ^{2}-4 \beta }\, \beta r -2 \sqrt {\alpha ^{2}-4 \beta }\, \beta +\sqrt {\alpha ^{2}-4 \beta }\, \delta -\alpha \delta +2 \beta \gamma \right )}{2 \beta }+a_{k} \left (\beta \,k^{2}+2 \beta k r +\beta \,r^{2}-\beta k -\beta r +\delta k +\delta r +\epsilon \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \frac {r \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta r -2 \sqrt {\alpha ^{2}-4 \beta }\, \beta +\sqrt {\alpha ^{2}-4 \beta }\, \delta -\alpha \delta +2 \beta \gamma \right )}{2 \beta }=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, \frac {2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma }{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \frac {2 a_{k +1} \left (k +1+r \right ) \left (\left (k +r \right ) \beta +\frac {\delta }{2}\right ) \sqrt {\alpha ^{2}-4 \beta }+2 a_{k} \left (k +r \right ) \left (k +r -1\right ) \beta ^{2}+\left (2 \gamma \left (k +1+r \right ) a_{k +1}+2 a_{k} \left (\delta k +\delta r +\epsilon \right )\right ) \beta -a_{k +1} \alpha \delta \left (k +1+r \right )}{2 \beta }=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {2 \beta a_{k} \left (\beta \,k^{2}+2 \beta k r +\beta \,r^{2}-\beta k -\beta r +\delta k +\delta r +\epsilon \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \,k^{2}+4 \sqrt {\alpha ^{2}-4 \beta }\, \beta k r +2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \,r^{2}+2 \sqrt {\alpha ^{2}-4 \beta }\, \beta k +2 \sqrt {\alpha ^{2}-4 \beta }\, \beta r +\sqrt {\alpha ^{2}-4 \beta }\, \delta k +\sqrt {\alpha ^{2}-4 \beta }\, \delta r -\alpha \delta k -\alpha \delta r +2 \beta \gamma k +2 \beta \gamma r +\sqrt {\alpha ^{2}-4 \beta }\, \delta -\alpha \delta +2 \beta \gamma } \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +1}=-\frac {2 \beta a_{k} \left (\beta \,k^{2}-\beta k +\delta k +\epsilon \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \,k^{2}+2 \sqrt {\alpha ^{2}-4 \beta }\, \beta k +\sqrt {\alpha ^{2}-4 \beta }\, \delta k -\alpha \delta k +2 \beta \gamma k +\sqrt {\alpha ^{2}-4 \beta }\, \delta -\alpha \delta +2 \beta \gamma } \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k}, a_{k +1}=-\frac {2 \beta a_{k} \left (\beta \,k^{2}-\beta k +\delta k +\epsilon \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \,k^{2}+2 \sqrt {\alpha ^{2}-4 \beta }\, \beta k +\sqrt {\alpha ^{2}-4 \beta }\, \delta k -\alpha \delta k +2 \beta \gamma k +\sqrt {\alpha ^{2}-4 \beta }\, \delta -\alpha \delta +2 \beta \gamma }\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta } \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} {\left (x -\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }\right )}^{k}, a_{k +1}=-\frac {2 \beta a_{k} \left (\beta \,k^{2}-\beta k +\delta k +\epsilon \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \,k^{2}+2 \sqrt {\alpha ^{2}-4 \beta }\, \beta k +\sqrt {\alpha ^{2}-4 \beta }\, \delta k -\alpha \delta k +2 \beta \gamma k +\sqrt {\alpha ^{2}-4 \beta }\, \delta -\alpha \delta +2 \beta \gamma }\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma }{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta } \\ {} & {} & a_{k +1}=-\frac {2 \beta a_{k} \left (\beta \,k^{2}+\frac {k \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{\sqrt {\alpha ^{2}-4 \beta }}+\frac {\left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )^{2}}{4 \beta \left (\alpha ^{2}-4 \beta \right )}-\beta k -\frac {2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma }{2 \sqrt {\alpha ^{2}-4 \beta }}+\delta k +\frac {\delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+\epsilon \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \,k^{2}+2 k \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )+\frac {\left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )^{2}}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+2 \sqrt {\alpha ^{2}-4 \beta }\, \beta k +2 \sqrt {\alpha ^{2}-4 \beta }\, \beta +\sqrt {\alpha ^{2}-4 \beta }\, \delta k +\frac {\delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \beta }-\alpha \delta k -\frac {\alpha \delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+2 \beta \gamma k +\frac {\gamma \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{\sqrt {\alpha ^{2}-4 \beta }}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma }{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta } \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma }{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }}, a_{k +1}=-\frac {2 \beta a_{k} \left (\beta \,k^{2}+\frac {k \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{\sqrt {\alpha ^{2}-4 \beta }}+\frac {\left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )^{2}}{4 \beta \left (\alpha ^{2}-4 \beta \right )}-\beta k -\frac {2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma }{2 \sqrt {\alpha ^{2}-4 \beta }}+\delta k +\frac {\delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+\epsilon \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \,k^{2}+2 k \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )+\frac {\left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )^{2}}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+2 \sqrt {\alpha ^{2}-4 \beta }\, \beta k +2 \sqrt {\alpha ^{2}-4 \beta }\, \beta +\sqrt {\alpha ^{2}-4 \beta }\, \delta k +\frac {\delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \beta }-\alpha \delta k -\frac {\alpha \delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+2 \beta \gamma k +\frac {\gamma \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{\sqrt {\alpha ^{2}-4 \beta }}}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta } \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} {\left (x -\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }\right )}^{k +\frac {2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma }{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }}, a_{k +1}=-\frac {2 \beta a_{k} \left (\beta \,k^{2}+\frac {k \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{\sqrt {\alpha ^{2}-4 \beta }}+\frac {\left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )^{2}}{4 \beta \left (\alpha ^{2}-4 \beta \right )}-\beta k -\frac {2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma }{2 \sqrt {\alpha ^{2}-4 \beta }}+\delta k +\frac {\delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+\epsilon \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \,k^{2}+2 k \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )+\frac {\left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )^{2}}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+2 \sqrt {\alpha ^{2}-4 \beta }\, \beta k +2 \sqrt {\alpha ^{2}-4 \beta }\, \beta +\sqrt {\alpha ^{2}-4 \beta }\, \delta k +\frac {\delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \beta }-\alpha \delta k -\frac {\alpha \delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+2 \beta \gamma k +\frac {\gamma \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{\sqrt {\alpha ^{2}-4 \beta }}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} {\left (x -\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }\right )}^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} {\left (x -\frac {-\alpha +\sqrt {\alpha ^{2}-4 \beta }}{2 \beta }\right )}^{k +\frac {2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma }{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }}\right ), a_{1+k}=-\frac {2 \beta a_{k} \left (\beta \,k^{2}-\beta k +\delta k +\epsilon \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \,k^{2}+2 \sqrt {\alpha ^{2}-4 \beta }\, \beta k +\sqrt {\alpha ^{2}-4 \beta }\, \delta k -\alpha \delta k +2 \beta \gamma k +\sqrt {\alpha ^{2}-4 \beta }\, \delta -\alpha \delta +2 \beta \gamma }, b_{1+k}=-\frac {2 \beta b_{k} \left (\beta \,k^{2}+\frac {k \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{\sqrt {\alpha ^{2}-4 \beta }}+\frac {\left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )^{2}}{4 \beta \left (\alpha ^{2}-4 \beta \right )}-\beta k -\frac {2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma }{2 \sqrt {\alpha ^{2}-4 \beta }}+\delta k +\frac {\delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+\epsilon \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta \,k^{2}+2 k \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )+\frac {\left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )^{2}}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+2 \sqrt {\alpha ^{2}-4 \beta }\, \beta k +2 \sqrt {\alpha ^{2}-4 \beta }\, \beta +\sqrt {\alpha ^{2}-4 \beta }\, \delta k +\frac {\delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \beta }-\alpha \delta k -\frac {\alpha \delta \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{2 \sqrt {\alpha ^{2}-4 \beta }\, \beta }+2 \beta \gamma k +\frac {\gamma \left (2 \sqrt {\alpha ^{2}-4 \beta }\, \beta -\sqrt {\alpha ^{2}-4 \beta }\, \delta +\alpha \delta -2 \beta \gamma \right )}{\sqrt {\alpha ^{2}-4 \beta }}}\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 -> hypergeometric -> heuristic approach <- heuristic approach successful <- hypergeometric successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 366
Order:=6; dsolve((1+alpha*x+beta*x^2)*diff(y(x),x$2)+(gamma+delta*x)*diff(y(x),x)+epsilon*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (1-\frac {\epsilon \,x^{2}}{2}+\frac {\epsilon \left (\alpha +\gamma \right ) x^{3}}{6}+\frac {\epsilon \left (-\alpha ^{2}-\frac {3}{2} \alpha \gamma -\frac {1}{2} \gamma ^{2}+\beta +\delta +\frac {1}{2} \epsilon \right ) x^{4}}{12}-\frac {\epsilon \left (\frac {\left (\alpha +\frac {\gamma }{2}\right ) \epsilon }{3}-\frac {\gamma ^{3}}{12}-\frac {\alpha \,\gamma ^{2}}{2}+\frac {\left (-\frac {11 \alpha ^{2}}{4}+2 \beta +\frac {5 \delta }{4}\right ) \gamma }{3}+\alpha \left (-\frac {\alpha ^{2}}{2}+\beta +\frac {3 \delta }{4}\right )\right ) x^{5}}{10}\right ) y \left (0\right )+\left (x -\frac {\gamma \,x^{2}}{2}+\frac {\left (\alpha \gamma +\gamma ^{2}-\delta -\epsilon \right ) x^{3}}{6}+\frac {\left (\left (2 \alpha +2 \gamma \right ) \epsilon -\gamma ^{3}-3 \alpha \,\gamma ^{2}+\left (-2 \alpha ^{2}+2 \beta +3 \delta \right ) \gamma +2 \delta \alpha \right ) x^{4}}{24}+\frac {\left (\epsilon ^{2}+\left (-6 \alpha ^{2}-9 \alpha \gamma -3 \gamma ^{2}+6 \beta +4 \delta \right ) \epsilon +\gamma ^{4}+6 \alpha \,\gamma ^{3}+\left (11 \alpha ^{2}-8 \beta -6 \delta \right ) \gamma ^{2}-12 \alpha \left (-\frac {\alpha ^{2}}{2}+\beta +\frac {7 \delta }{6}\right ) \gamma +6 \left (-\alpha ^{2}+\beta +\frac {\delta }{2}\right ) \delta \right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 561
AsymptoticDSolveValue[(1+\[Alpha]*x+\[Beta]*x^2)*y''[x]+(\[Gamma]+\[Delta]*x)*y'[x]+\[Epsilon]*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {1}{20} x^5 \epsilon \left (2 \alpha \beta -\alpha ^3\right )-\frac {1}{20} x^5 \epsilon \left (\alpha \delta -\gamma \left (\alpha ^2-\beta \right )\right )+\frac {1}{60} \gamma x^5 \epsilon \left (\alpha ^2-\beta \right )+\frac {1}{120} \alpha \gamma ^2 x^5 \epsilon +\frac {1}{40} \alpha x^5 \epsilon (\alpha \gamma -\delta )+\frac {1}{24} \gamma x^5 \epsilon (\alpha \gamma -\delta )-\frac {1}{30} \alpha x^5 \epsilon ^2+\frac {1}{120} \gamma ^3 x^5 \epsilon -\frac {1}{60} \gamma x^5 \epsilon ^2-\frac {1}{12} x^4 \epsilon \left (\alpha ^2-\beta \right )-\frac {1}{12} x^4 \epsilon (\alpha \gamma -\delta )-\frac {1}{24} \alpha \gamma x^4 \epsilon -\frac {1}{24} \gamma ^2 x^4 \epsilon +\frac {x^4 \epsilon ^2}{24}+\frac {1}{6} \alpha x^3 \epsilon +\frac {1}{6} \gamma x^3 \epsilon -\frac {x^2 \epsilon }{2}+1\right )+c_2 \left (\frac {1}{60} \gamma x^5 \left (\gamma \left (\alpha ^2-\beta \right )-\alpha \delta \right )-\frac {1}{20} \gamma x^5 \left (\alpha \delta -\gamma \left (\alpha ^2-\beta \right )\right )-\frac {1}{20} x^5 \epsilon \left (\alpha ^2-\beta \right )-\frac {1}{20} x^5 \left (\alpha ^3 (-\gamma )+\alpha ^2 \delta +2 \alpha \beta \gamma -\beta \delta \right )+\frac {1}{24} \gamma ^2 x^5 (\alpha \gamma -\delta )-\frac {1}{120} \gamma ^2 x^5 (\delta -\alpha \gamma )-\frac {1}{40} x^5 \epsilon (\alpha \gamma -\delta )+\frac {1}{120} x^5 \epsilon (\delta -\alpha \gamma )-\frac {1}{40} x^5 (\alpha \gamma -\delta ) (\delta -\alpha \gamma )-\frac {1}{24} \alpha \gamma x^5 \epsilon +\frac {\gamma ^4 x^5}{120}-\frac {1}{40} \gamma ^2 x^5 \epsilon +\frac {x^5 \epsilon ^2}{120}-\frac {1}{12} x^4 \left (\gamma \left (\alpha ^2-\beta \right )-\alpha \delta \right )-\frac {1}{12} \gamma x^4 (\alpha \gamma -\delta )+\frac {1}{24} \gamma x^4 (\delta -\alpha \gamma )+\frac {1}{12} \alpha x^4 \epsilon -\frac {\gamma ^3 x^4}{24}+\frac {1}{12} \gamma x^4 \epsilon -\frac {1}{6} x^3 (\delta -\alpha \gamma )+\frac {\gamma ^2 x^3}{6}-\frac {x^3 \epsilon }{6}-\frac {\gamma x^2}{2}+x\right ) \]