59.1.582 problem 598

Internal problem ID [9754]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 598
Date solved : Sunday, March 30, 2025 at 02:45:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

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

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

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

✓ Mathematica. Time used: 0.273 (sec). Leaf size: 47

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

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

✗ Sympy

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

False