59.1.692 problem 709

Internal problem ID [9864]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 709
Date solved : Sunday, March 30, 2025 at 02:48:23 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4}&=0 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 32

ode:=x*(1-x)*diff(diff(y(x),x),x)+(3/2-2*x)*diff(y(x),x)-1/4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_2 \ln \left (-1+2 x +2 \sqrt {x \left (-1+x \right )}\right )-c_2 \ln \left (2\right )+c_1}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.273 (sec). Leaf size: 104

ode=x*(1-x)*D[y[x],{x,2}]+(3/2-2*x)*D[y[x],x]-1/4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {1}{4} \left (\frac {1}{K[1]}+\frac {1}{K[1]-1}\right )dK[1]-\frac {1}{2} \int _1^x\frac {1}{2} \left (\frac {3}{K[2]}+\frac {1}{K[2]-1}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {1-2 K[1]}{4 K[1]-4 K[1]^2}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x)*Derivative(y(x), (x, 2)) + (3/2 - 2*x)*Derivative(y(x), x) - y(x)/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False