59.1.150 problem 152

Internal problem ID [9322]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 152
Date solved : Sunday, March 30, 2025 at 02:32:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

\[ y = c_2 \left (-1+x \right ) \ln \left (x \right )-4 c_2 +c_1 \left (-1+x \right ) \]

✓ Mathematica. Time used: 0.451 (sec). Leaf size: 112

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

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

✗ Sympy

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

False