54.3.236 problem 1252

Internal problem ID [12531]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1252
Date solved : Friday, October 03, 2025 at 03:30:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 124

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

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

✓ Mathematica. Time used: 0.129 (sec). Leaf size: 131

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

\begin{align*} y(x)&\to c_2 x^{1-b} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (a-2 b-\sqrt {a^2-2 a-4 c+1}+1\right ),\frac {1}{2} \left (a-2 b+\sqrt {a^2-2 a-4 c+1}+1\right ),2-b,-x\right )+c_1 \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 ),b,-x\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*(x + 1)*Derivative(y(x), (x, 2)) + (a*x + b)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None