59.1.268 problem 271

Internal problem ID [9440]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 271
Date solved : Wednesday, March 05, 2025 at 07:49:17 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 25

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

\[ y = c_{1} x +\arctan \left (\sqrt {x -1}\right ) x c_{2} -\sqrt {x -1}\, c_{2} \]

✓ Mathematica. Time used: 0.422 (sec). Leaf size: 75

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

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

✗ Sympy

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

False