59.1.695 problem 712

Internal problem ID [9867]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 712
Date solved : Sunday, March 30, 2025 at 02:48:27 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9}&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 27

ode:=x*(1-x)*diff(diff(y(x),x),x)+1/3*(-2*x+1)*diff(y(x),x)+20/9*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \left (-5+6 x \right ) x^{{2}/{3}}+c_2 \left (6 x -1\right ) \left (-1+x \right )^{{2}/{3}} \]

✓ Mathematica. Time used: 0.494 (sec). Leaf size: 93

ode=x*(1-x)*D[y[x],{x,2}]+1/3*(1-2*x)*D[y[x],x]+20/9*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\left (3 c_2 (x-1) \operatorname {Gamma}\left (\frac {4}{3}\right ) Q_1^{\frac {2}{3}}(2 x-1)+c_1 (5-6 x) (1-x)^{2/3} \sqrt [3]{x}\right ) \exp \left (\int _1^x\frac {1-2 K[1]}{3 K[1]-3 K[1]^2}dK[1]\right )}{3 (x-1) \operatorname {Gamma}\left (\frac {4}{3}\right )} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x)*Derivative(y(x), (x, 2)) + (1 - 2*x)*Derivative(y(x), x)/3 + 20*y(x)/9,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False