Internal problem ID [10962]
Internal file name [OUTPUT/10219_Sunday_December_31_2023_11_10_12_AM_23575013/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 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 139.
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.
\[ \boxed {a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & a_{2} x^{2} \left (\frac {d}{d x}y^{\prime }\right )+x \left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) 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 {\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y}{a_{2} x^{2}}-\frac {\left (a_{1} x +b_{1} \right ) y^{\prime }}{x a_{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 (a_{1} x +b_{1} \right ) y^{\prime }}{x a_{2}}+\frac {\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y}{a_{2} x^{2}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {a_{1} x +b_{1}}{a_{2} x}, P_{3}\left (x \right )=\frac {a_{0} x^{2}+b_{0} x +c_{0}}{a_{2} x^{2}}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=\frac {b_{1}}{a_{2}} \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=\frac {c_{0}}{a_{2}} \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & a_{2} x^{2} \left (\frac {d}{d x}y^{\prime }\right )+x \left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (a_{2} r^{2}-a_{2} r +b_{1} r +c_{0} \right ) x^{r}+\left (\left (a_{2} r^{2}+a_{2} r +b_{1} r +b_{1} +c_{0} \right ) a_{1}+a_{0} \left (a_{1} r +b_{0} \right )\right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (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 )+a_{k -1} \left (a_{1} \left (k -1\right )+a_{1} r +b_{0} \right )+a_{k -2} a_{0} \right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & a_{2} r^{2}-a_{2} r +b_{1} r +c_{0} =0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{\frac {a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2 a_{2}}, -\frac {-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2 a_{2}}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & \left (a_{2} r^{2}+a_{2} r +b_{1} r +b_{1} +c_{0} \right ) a_{1}+a_{0} \left (a_{1} r +b_{0} \right )=0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=-\frac {a_{0} \left (a_{1} r +b_{0} \right )}{a_{2} r^{2}+a_{2} r +b_{1} r +b_{1} +c_{0}} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (\left (k +r \right ) \left (k +r -1\right ) a_{2} +b_{1} k +b_{1} r +c_{0} \right ) a_{k}+a_{1} k a_{k -1}+a_{1} r a_{k -1}+\left (-a_{1} +b_{0} \right ) a_{k -1}+a_{k -2} a_{0} =0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (\left (k +2+r \right ) \left (k +1+r \right ) a_{2} +b_{1} \left (k +2\right )+b_{1} r +c_{0} \right ) a_{k +2}+a_{1} \left (k +2\right ) a_{k +1}+a_{1} r a_{k +1}+\left (-a_{1} +b_{0} \right ) a_{k +1}+a_{k} a_{0} =0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {a_{1} k a_{k +1}+a_{1} r a_{k +1}+a_{k} a_{0} +a_{1} a_{k +1}+b_{0} a_{k +1}}{a_{2} k^{2}+2 a_{2} k r +a_{2} r^{2}+3 a_{2} k +3 a_{2} r +b_{1} k +b_{1} r +2 a_{2} +2 b_{1} +c_{0}} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2 a_{2}} \\ {} & {} & a_{k +2}=-\frac {a_{1} k a_{k +1}+\frac {a_{1} \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right ) a_{k +1}}{2 a_{2}}+a_{k} a_{0} +a_{1} a_{k +1}+b_{0} a_{k +1}}{a_{2} k^{2}+k \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )+\frac {\left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )^{2}}{4 a_{2}}+3 a_{2} k +\frac {7 a_{2}}{2}+\frac {b_{1}}{2}+\frac {3 \sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2}+b_{1} k +\frac {b_{1} \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+c_{0}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2 a_{2}} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2 a_{2}}}, a_{k +2}=-\frac {a_{1} k a_{k +1}+\frac {a_{1} \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right ) a_{k +1}}{2 a_{2}}+a_{k} a_{0} +a_{1} a_{k +1}+b_{0} a_{k +1}}{a_{2} k^{2}+k \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )+\frac {\left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )^{2}}{4 a_{2}}+3 a_{2} k +\frac {7 a_{2}}{2}+\frac {b_{1}}{2}+\frac {3 \sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2}+b_{1} k +\frac {b_{1} \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+c_{0}}, a_{1}=-\frac {a_{0} \left (\frac {a_{1} \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+b_{0} \right )}{\frac {\left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )^{2}}{4 a_{2}}+\frac {a_{2}}{2}+\frac {b_{1}}{2}+\frac {\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2}+\frac {b_{1} \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+c_{0}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2 a_{2}} \\ {} & {} & a_{k +2}=-\frac {a_{1} k a_{k +1}-\frac {a_{1} \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right ) a_{k +1}}{2 a_{2}}+a_{k} a_{0} +a_{1} a_{k +1}+b_{0} a_{k +1}}{a_{2} k^{2}-k \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )+\frac {\left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )^{2}}{4 a_{2}}+3 a_{2} k +\frac {7 a_{2}}{2}+\frac {b_{1}}{2}-\frac {3 \sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2}+b_{1} k -\frac {b_{1} \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+c_{0}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2 a_{2}} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2 a_{2}}}, a_{k +2}=-\frac {a_{1} k a_{k +1}-\frac {a_{1} \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right ) a_{k +1}}{2 a_{2}}+a_{k} a_{0} +a_{1} a_{k +1}+b_{0} a_{k +1}}{a_{2} k^{2}-k \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )+\frac {\left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )^{2}}{4 a_{2}}+3 a_{2} k +\frac {7 a_{2}}{2}+\frac {b_{1}}{2}-\frac {3 \sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2}+b_{1} k -\frac {b_{1} \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+c_{0}}, a_{1}=-\frac {a_{0} \left (-\frac {a_{1} \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+b_{0} \right )}{\frac {\left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )^{2}}{4 a_{2}}+\frac {a_{2}}{2}+\frac {b_{1}}{2}-\frac {\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2}-\frac {b_{1} \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+c_{0}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2 a_{2}}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k -\frac {-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2 a_{2}}}\right ), a_{k +2}=-\frac {a_{1} k a_{k +1}+\frac {a_{1} \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right ) a_{k +1}}{2 a_{2}}+a_{k} a_{0} +a_{1} a_{k +1}+b_{0} a_{k +1}}{a_{2} k^{2}+k \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )+\frac {\left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )^{2}}{4 a_{2}}+3 a_{2} k +\frac {7 a_{2}}{2}+\frac {b_{1}}{2}+\frac {3 \sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2}+b_{1} k +\frac {b_{1} \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+c_{0}}, a_{1}=-\frac {a_{0} \left (\frac {a_{1} \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+b_{0} \right )}{\frac {\left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )^{2}}{4 a_{2}}+\frac {a_{2}}{2}+\frac {b_{1}}{2}+\frac {\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2}+\frac {b_{1} \left (a_{2} -b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+c_{0}}, b_{k +2}=-\frac {a_{1} k b_{k +1}-\frac {a_{1} \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right ) b_{k +1}}{2 a_{2}}+b_{k} a_{0} +a_{1} b_{k +1}+b_{0} b_{k +1}}{a_{2} k^{2}-k \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )+\frac {\left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )^{2}}{4 a_{2}}+3 a_{2} k +\frac {7 a_{2}}{2}+\frac {b_{1}}{2}-\frac {3 \sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2}+b_{1} k -\frac {b_{1} \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+c_{0}}, b_{1}=-\frac {b_{0} \left (-\frac {a_{1} \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+b_{0} \right )}{\frac {\left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )^{2}}{4 a_{2}}+\frac {a_{2}}{2}+\frac {b_{1}}{2}-\frac {\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}}{2}-\frac {b_{1} \left (-a_{2} +b_{1} +\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 c_{0} a_{2} +b_{1}^{2}}\right )}{2 a_{2}}+c_{0}}\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 -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Whittaker successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 150
dsolve(a__2*x^2*diff(y(x),x$2)+(a__1*x^2+b__1*x)*diff(y(x),x)+(a__0*x^2+b__0*x+c__0)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {a_{1} x}{2 a_{2}}} x^{-\frac {b_{1}}{2 a_{2}}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {b_{1} a_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right )+\operatorname {WhittakerW}\left (-\frac {b_{1} a_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.538 (sec). Leaf size: 272
DSolve[a2*x^2*y''[x]+(a1*x^2+b1*x)*y'[x]+(a0*x^2+b0*x+c0)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {x \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}\right )}{2 \text {a2}}} x^{\frac {\sqrt {\text {a2}^2-2 \text {a2} (\text {b1}+2 \text {c0})+\text {b1}^2}+\text {a2}-\text {b1}}{2 \text {a2}}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}},\frac {\text {a2}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}},\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )+c_2 L_{-\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}}}^{\frac {\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}}}\left (\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )\right ) \]