2.31.18 Problem 166

Internal problem ID [13827]
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
Problem number : 166
Date solved : Friday, December 19, 2025 at 02:01:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (1+2 n \right ) a x y^{\prime }+y c&=0 \\ \end{align*}

2.31.18.1 ✓ Maple. Time used: 0.016 (sec). Leaf size: 93

ode:=(a*x^2+b)*diff(diff(y(x),x),x)+(1+2*n)*a*x*diff(y(x),x)+c*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (a \,x^{2}+b \right )^{\frac {1}{4}-\frac {n}{2}} \left (\operatorname {LegendreQ}\left (-\frac {-2 \sqrt {a \,n^{2}-c}+\sqrt {a}}{2 \sqrt {a}}, -\frac {1}{2}+n , \frac {a x}{\sqrt {-a b}}\right ) c_2 +\operatorname {LegendreP}\left (-\frac {-2 \sqrt {a \,n^{2}-c}+\sqrt {a}}{2 \sqrt {a}}, -\frac {1}{2}+n , \frac {a x}{\sqrt {-a b}}\right ) c_1 \right ) \]

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
 

2.31.18.2 ✓ Mathematica. Time used: 0.053 (sec). Leaf size: 118

ode=(a*x^2+b)*D[y[x],{x,2}]+(2*n+1)*a*x*D[y[x],x]+c*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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 ) \end{align*}

2.31.18.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a*x*(2*n + 1)*Derivative(y(x), x) + c*y(x) + (a*x**2 + b)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False