54.3.233 problem 1249

Internal problem ID [12528]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1249
Date solved : Friday, October 03, 2025 at 03:30:45 AM
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: 0.025 (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);
 

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

✓ Mathematica. Time used: 0.127 (sec). Leaf size: 190

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

\begin{align*} y(x)&\to \frac {1}{2} (x-1)^{\frac {1}{2} (-a-b)} \left (2 c_1 (x-1)^{\frac {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 {a+b}{2},\frac {1-x}{2}\right )+c_2 (x-1) 2^{\frac {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 )\right ) \end{align*}

✗ Sympy

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

False