61.30.10 problem 158

Internal problem ID [12579]
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 : 158
Date solved : Monday, March 31, 2025 at 05:39:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 112

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

\[ y = c_1 \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {8 b a +4 b^{2}+1}}{2}, \frac {\sqrt {8 b a +4 b^{2}+1}}{2}-\frac {1}{2}\right ], \left [-a -\frac {1}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )+c_2 \left (\frac {x}{2}+\frac {1}{2}\right )^{\frac {3}{2}+a} \operatorname {hypergeom}\left (\left [1-\frac {\sqrt {8 b a +4 b^{2}+1}}{2}+a , \frac {\sqrt {8 b a +4 b^{2}+1}}{2}+1+a \right ], \left [\frac {5}{2}+a \right ], \frac {x}{2}+\frac {1}{2}\right ) \]

✓ Mathematica. Time used: 0.185 (sec). Leaf size: 152

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

\[ y(x)\to 2^{a-\frac {1}{2}} c_2 (x-1)^{\frac {1}{2}-a} \operatorname {Hypergeometric2F1}\left (-a-\frac {1}{2} \sqrt {4 b^2+8 a b+1},\frac {1}{2} \sqrt {4 b^2+8 a b+1}-a,\frac {3}{2}-a,\frac {1}{2}-\frac {x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-\sqrt {4 b^2+8 a b+1}-1\right ),\frac {1}{2} \left (\sqrt {4 b^2+8 a b+1}-1\right ),a+\frac {1}{2},\frac {1-x}{2}\right ) \]

✗ Sympy

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

False