59.1.242 problem 245

Internal problem ID [9414]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 245
Date solved : Sunday, March 30, 2025 at 02:34:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 17

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

\[ y = \frac {c_2 \,{\mathrm e}^{x}+c_1 x +c_1}{x} \]

✓ Mathematica. Time used: 0.282 (sec). Leaf size: 80

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

\begin{align*} y(x)\to \exp \left (\int _1^x\left (1-\frac {1}{K[1]}\right )dK[1]\right ) \left (\int _1^x\exp \left (-\int _1^{K[2]}\left (1-\frac {1}{K[1]}\right )dK[1]\right ) c_1dK[2]+c_2\right ) \\ y(x)\to c_2 \exp \left (\int _1^x\left (1-\frac {1}{K[1]}\right )dK[1]\right ) \\ \end{align*}

✓ Sympy. Time used: 1.066 (sec). Leaf size: 386

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) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]