61.30.20 problem 168

Internal problem ID [12589]
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 : 168
Date solved : Thursday, March 13, 2025 at 11:52:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.306 (sec). Leaf size: 134

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

\[ y = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {a}{2}-\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}, -\frac {1}{2}-\frac {a}{2}+\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}\right ], \left [-\frac {a}{2}+\frac {b}{2}\right ], \frac {1}{2}+\frac {x}{2}\right )+c_{2} \left (\frac {1}{2}+\frac {x}{2}\right )^{1+\frac {a}{2}-\frac {b}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}-\frac {b}{2}, \frac {1}{2}+\frac {\sqrt {a^{2}+2 a +4 c +1}}{2}-\frac {b}{2}\right ], \left [2+\frac {a}{2}-\frac {b}{2}\right ], \frac {1}{2}+\frac {x}{2}\right ) \]

✓ Mathematica. Time used: 0.194 (sec). Leaf size: 184

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

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

✗ Sympy

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

False