54.3.244 problem 1260

Internal problem ID [12539]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1260
Date solved : Wednesday, October 01, 2025 at 01:57:22 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x \left (x -1\right ) y^{\prime \prime }+\left (\left (\operatorname {a1} +\operatorname {b1} +1\right ) x -\operatorname {d1} \right ) y^{\prime }+\operatorname {a1} \operatorname {b1} \operatorname {d1}&=0 \end{align*}

✓ Maple. Time used: 0.135 (sec). Leaf size: 78

ode:=x*(x-1)*diff(diff(y(x),x),x)+((a1+b1+1)*x-d1)*diff(y(x),x)+a1*b1*d1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\int \left (x -1\right )^{\operatorname {d1} -\operatorname {a1} -\operatorname {b1} -1} \left (\operatorname {a1} \operatorname {b1} \operatorname {signum}\left (x -1\right )^{-\operatorname {d1} +\operatorname {a1} +\operatorname {b1}} \left (-\operatorname {signum}\left (x -1\right )\right )^{\operatorname {d1} -\operatorname {a1} -\operatorname {b1}} \operatorname {hypergeom}\left (\left [\operatorname {d1} , \operatorname {d1} -\operatorname {a1} -\operatorname {b1} \right ], \left [1+\operatorname {d1} \right ], x\right )-x^{-\operatorname {d1}} c_1 \right )d x +c_2 \]

✓ Mathematica. Time used: 1.908 (sec). Leaf size: 106

ode=a1*b1*d1 + (-d1 + (1 + a1 + b1)*x)*D[y[x],x] + (-1 + x)*x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\exp \left (\int _1^{K[3]}\frac {\text {d1}-(\text {a1}+\text {b1}+1) K[1]}{(K[1]-1) K[1]}dK[1]\right ) \left (c_1+\int _1^{K[3]}-\frac {\text {a1} \text {b1} \text {d1} \exp \left (-\int _1^{K[2]}\frac {\text {d1}-(\text {a1}+\text {b1}+1) K[1]}{(K[1]-1) K[1]}dK[1]\right )}{(K[2]-1) K[2]}dK[2]\right )dK[3]+c_2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a1 = symbols("a1") 
b1 = symbols("b1") 
d1 = symbols("d1") 
y = Function("y") 
ode = Eq(a1*b1*d1 + x*(x - 1)*Derivative(y(x), (x, 2)) + (-d1 + x*(a1 + b1 + 1))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out