Internal problem ID [10989]
Internal file name [OUTPUT/10246_Sunday_December_31_2023_11_14_37_AM_52742929/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: 166.
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.
\[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (2 n +1\right ) a x y^{\prime }+c y=0} \]
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 <- Legendre successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 93
dsolve((a*x^2+b)*diff(y(x),x$2)+(2*n+1)*a*x*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (a \,x^{2}+b \right )^{-\frac {n}{2}+\frac {1}{4}} \left (c_{1} \operatorname {LegendreP}\left (-\frac {-2 \sqrt {a \,n^{2}-c}+\sqrt {a}}{2 \sqrt {a}}, n -\frac {1}{2}, \frac {a x}{\sqrt {-a b}}\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {-2 \sqrt {a \,n^{2}-c}+\sqrt {a}}{2 \sqrt {a}}, n -\frac {1}{2}, \frac {a x}{\sqrt {-a b}}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.149 (sec). Leaf size: 118
DSolve[(a*x^2+b)*y''[x]+(2*n+1)*a*x*y'[x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (a x^2+b\right )^{\frac {1}{4}-\frac {n}{2}} \left (c_1 P_{\frac {\sqrt {a n^2-c}}{\sqrt {a}}-\frac {1}{2}}^{n-\frac {1}{2}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )+c_2 Q_{\frac {\sqrt {a n^2-c}}{\sqrt {a}}-\frac {1}{2}}^{n-\frac {1}{2}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )\right ) \]