61.30.18 problem 166

Internal problem ID [12587]
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 : Monday, March 31, 2025 at 05:43:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 93

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

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

✓ Mathematica. Time used: 0.089 (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]
 

\[ y(x)\to \left (a x^2+b\right )^{\frac {1}{4}-\frac {n}{2}} \left (c_1 P_{\frac {\sqrt {a n^2-c}}{\sqrt {a}}-\frac {1}{2}}^{n-\frac {1}{2}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )+c_2 Q_{\frac {\sqrt {a n^2-c}}{\sqrt {a}}-\frac {1}{2}}^{n-\frac {1}{2}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )\right ) \]

✗ 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