30.31 problem 179

Internal problem ID [11002]
Internal file name [OUTPUT/10259_Sunday_December_31_2023_11_33_17_AM_58603201/index.tex]

Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form \((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 179.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[[_2nd_order, _with_linear_symmetries]]

Unable to solve or complete the solution.

\[ \boxed {\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y=0} \]

30.31.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) \left (\frac {d}{d x}y^{\prime }\right )+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=-\frac {c_{0} y}{a_{2} x^{2}+b_{2} x +c_{2}}-\frac {\left (b_{1} x +c_{1} \right ) y^{\prime }}{a_{2} x^{2}+b_{2} x +c_{2}} \\ \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} \\ {} & {} & \frac {d}{d x}y^{\prime }+\frac {\left (b_{1} x +c_{1} \right ) y^{\prime }}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {c_{0} y}{a_{2} x^{2}+b_{2} x +c_{2}}=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 {b_{1} x +c_{1}}{a_{2} x^{2}+b_{2} x +c_{2}}, P_{3}\left (x \right )=\frac {c_{0}}{a_{2} x^{2}+b_{2} x +c_{2}}\right ] \\ {} & \circ & \left (x -\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}} \\ {} & {} & \left (\left (x -\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}}}}=0 \\ {} & \circ & {\left (x -\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}} \\ {} & {} & \left ({\left (x -\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}}}}=0 \\ {} & \circ & x =\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}\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 {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}} \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) \left (\frac {d}{d x}y^{\prime }\right )+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u +\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (a_{2} u^{2}+u \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\right ) \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )+\left (b_{1} u -\frac {b_{1} b_{2}}{2 a_{2}}+\frac {b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}+c_{1} \right ) \left (\frac {d}{d u}y \left (u \right )\right )+c_{0} 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}{d u}\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}\frac {d}{d u}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}{d u}\frac {d}{d u}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 {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} r -2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} +b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}+2 c_{1} a_{2} -b_{1} b_{2} \right ) u^{-1+r}}{2 a_{2}}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (\frac {a_{k +1} \left (k +1+r \right ) \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} \left (k +1\right )+2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} r -2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} +b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}+2 c_{1} a_{2} -b_{1} b_{2} \right )}{2 a_{2}}+a_{k} \left (a_{2} k^{2}+2 a_{2} k r +a_{2} r^{2}-a_{2} k -a_{2} r +b_{1} k +b_{1} r +c_{0} \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 {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} r -2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} +b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}+2 c_{1} a_{2} -b_{1} b_{2} \right )}{2 a_{2}}=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, \frac {2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \frac {2 \left (\left (k +r \right ) a_{2} +\frac {b_{1}}{2}\right ) a_{k +1} \left (k +1+r \right ) \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}+2 a_{k} \left (k +r \right ) \left (k +r -1\right ) a_{2}^{2}+\left (2 c_{1} \left (k +1+r \right ) a_{k +1}+2 a_{k} \left (b_{1} k +b_{1} r +c_{0} \right )\right ) a_{2} -b_{1} b_{2} a_{k +1} \left (k +1+r \right )}{2 a_{2}}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}+2 a_{2} k r +a_{2} r^{2}-a_{2} k -a_{2} r +b_{1} k +b_{1} r +c_{0} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k^{2}+4 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k r +2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} r^{2}+2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k +2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} r +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, b_{1} k +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, b_{1} r +2 c_{1} a_{2} k +2 c_{1} a_{2} r -b_{1} b_{2} k -b_{1} b_{2} r +b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}+2 c_{1} a_{2} -b_{1} b_{2}} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}-a_{2} k +b_{1} k +c_{0} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, b_{1} k +2 c_{1} a_{2} k -b_{1} b_{2} k +b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}+2 c_{1} a_{2} -b_{1} b_{2}} \\ \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 a_{2} a_{k} \left (a_{2} k^{2}-a_{2} k +b_{1} k +c_{0} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, b_{1} k +2 c_{1} a_{2} k -b_{1} b_{2} k +b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}+2 c_{1} a_{2} -b_{1} b_{2}}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} {\left (x -\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{k}, a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}-a_{2} k +b_{1} k +c_{0} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, b_{1} k +2 c_{1} a_{2} k -b_{1} b_{2} k +b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}+2 c_{1} a_{2} -b_{1} b_{2}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}} \\ {} & {} & a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}+\frac {k \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}+\frac {\left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )^{2}}{4 a_{2} \left (-4 c_{2} a_{2} +b_{2}^{2}\right )}-a_{2} k -\frac {2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}+b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}+c_{0} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 k \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )+\frac {\left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )^{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}+2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k +2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 a_{2}}+2 c_{1} a_{2} k +\frac {c_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}-b_{1} b_{2} k -\frac {b_{1} b_{2} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}}, a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}+\frac {k \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}+\frac {\left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )^{2}}{4 a_{2} \left (-4 c_{2} a_{2} +b_{2}^{2}\right )}-a_{2} k -\frac {2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}+b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}+c_{0} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 k \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )+\frac {\left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )^{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}+2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k +2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 a_{2}}+2 c_{1} a_{2} k +\frac {c_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}-b_{1} b_{2} k -\frac {b_{1} b_{2} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} {\left (x -\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{k +\frac {2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}}, a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}+\frac {k \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}+\frac {\left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )^{2}}{4 a_{2} \left (-4 c_{2} a_{2} +b_{2}^{2}\right )}-a_{2} k -\frac {2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}+b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}+c_{0} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 k \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )+\frac {\left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )^{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}+2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k +2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 a_{2}}+2 c_{1} a_{2} k +\frac {c_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}-b_{1} b_{2} k -\frac {b_{1} b_{2} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} {\left (x -\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} {\left (x -\frac {-b_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}{2 a_{2}}\right )}^{k +\frac {2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}}\right ), a_{k +1}=-\frac {2 a_{2} a_{k} \left (a_{2} k^{2}-a_{2} k +b_{1} k +c_{0} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, b_{1} k +2 c_{1} a_{2} k -b_{1} b_{2} k +b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}+2 c_{1} a_{2} -b_{1} b_{2}}, b_{k +1}=-\frac {2 a_{2} b_{k} \left (a_{2} k^{2}+\frac {k \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}+\frac {\left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )^{2}}{4 a_{2} \left (-4 c_{2} a_{2} +b_{2}^{2}\right )}-a_{2} k -\frac {2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}+b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}+c_{0} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k^{2}+2 k \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )+\frac {\left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )^{2}}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}+2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} k +2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} +\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, b_{1} k +\frac {b_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 a_{2}}+2 c_{1} a_{2} k +\frac {c_{1} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{\sqrt {-4 c_{2} a_{2} +b_{2}^{2}}}-b_{1} b_{2} k -\frac {b_{1} b_{2} \left (2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2} -b_{1} \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}-2 c_{1} a_{2} +b_{1} b_{2} \right )}{2 \sqrt {-4 c_{2} a_{2} +b_{2}^{2}}\, a_{2}}}\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: 0.14 (sec). Leaf size: 482

dsolve((a__2*x^2+b__2*x+c__2)*diff(y(x),x$2)+(b__1*x+c__1)*diff(y(x),x)+c__0*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{3} \operatorname {hypergeom}\left (\left [\frac {-a_{2} +b_{1} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, -\frac {a_{2} -b_{1} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}\right ], \left [\frac {b_{1} \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2} -2 a_{2} c_{1} +b_{1} b_{2}}{2 a_{2}^{2} \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}}\right ], \frac {\left (-2 x \,a_{2}^{2}-b_{2} a_{2} \right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 c_{2} a_{2} -b_{2}^{2}}{8 c_{2} a_{2} -2 b_{2}^{2}}\right )+c_{4} {\left (2 \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, x \,a_{2}^{2}+\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, b_{2} a_{2} -4 c_{2} a_{2} +b_{2}^{2}\right )}^{\frac {a_{2} \left (a_{2} -\frac {b_{1}}{2}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}} \operatorname {hypergeom}\left (\left [\frac {\frac {a_{2} \left (a_{2} -\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}}{2}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}, \frac {\frac {a_{2} \left (a_{2} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}}{2}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}\right ], \left [\frac {2 a_{2} \left (a_{2} -\frac {b_{1}}{4}\right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}\right ], \frac {\left (-2 x \,a_{2}^{2}-b_{2} a_{2} \right ) \sqrt {\frac {-4 c_{2} a_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 c_{2} a_{2} -b_{2}^{2}}{8 c_{2} a_{2} -2 b_{2}^{2}}\right ) \]

✓ Solution by Mathematica

Time used: 6.771 (sec). Leaf size: 498

DSolve[(a2*x^2+b2*x+c2)*y''[x]+(b1*x+c1)*y'[x]+c0*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}\left (-\frac {\text {a2}-\text {b1}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}}{2 \text {a2}},\frac {-\text {a2}+\text {b1}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}}{2 \text {a2}},\frac {\text {b1} \left (\text {b2}+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}\right )-2 \text {a2} \text {c1}}{2 \text {a2} \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}},\frac {\text {b2}+2 \text {a2} x+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}{2 \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right )-c_2 2^{\frac {\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}}{2 \text {a2}}-\frac {\text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}-1} \exp \left (-\frac {i \pi \left (\text {b1} \left (\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}+\text {b2}\right )-2 \text {a2} \text {c1}\right )}{2 \text {a2} \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right ) \left (\frac {\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}+2 \text {a2} x+\text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right )^{-\frac {\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}}{2 \text {a2}}+\frac {\text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+1} \operatorname {Hypergeometric2F1}\left (\frac {\frac {2 \text {c1} \text {a2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {a2}-\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}-\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}}{2 \text {a2}},\frac {\frac {2 \text {c1} \text {a2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {a2}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}-\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}}{2 \text {a2}},-\frac {\frac {\text {b2} \text {b1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}+\text {a2} \left (-\frac {2 \text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}-4\right )}{2 \text {a2}},\frac {\text {b2}+2 \text {a2} x+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}{2 \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right ) \]