59.1.345 problem 352

Internal problem ID [9517]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 352
Date solved : Sunday, March 30, 2025 at 02:36:38 PM
CAS classification : [_Jacobi]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 53

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

\[ y = \left (-12 x -3\right ) c_2 \ln \left (-1+2 x +2 \sqrt {x \left (-1+x \right )}\right )+\left (4 x +26\right ) c_2 \sqrt {x \left (-1+x \right )}+4 \left (3 \ln \left (2\right ) c_2 +c_1 \right ) \left (x +\frac {1}{4}\right ) \]

✓ Mathematica. Time used: 0.532 (sec). Leaf size: 130

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

\[ y(x)\to \frac {1}{4} (4 x+1) \exp \left (\int _1^x\frac {4 K[1]+1}{4 K[1]-4 K[1]^2}dK[1]-\frac {1}{2} \int _1^x\frac {4 K[2]+1}{2 K[2]-2 K[2]^2}dK[2]\right ) \left (c_2 \int _1^x\frac {16 \exp \left (-2 \int _1^{K[3]}\frac {4 K[1]+1}{4 K[1]-4 K[1]^2}dK[1]\right )}{(4 K[3]+1)^2}dK[3]+c_1\right ) \]

✗ Sympy

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

False