59.1.249 problem 252

Internal problem ID [9421]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 252
Date solved : Sunday, March 30, 2025 at 02:34:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 13

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

\[ y = x \left (\operatorname {Ei}_{1}\left (x \right ) c_2 +c_1 \right ) \]

✓ Mathematica. Time used: 0.443 (sec). Leaf size: 63

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

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

✗ Sympy

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

False