59.1.135 problem 137

Internal problem ID [9307]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 137
Date solved : Sunday, March 30, 2025 at 02:32:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-x \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 52

ode:=2*x^2*(1+x)*diff(diff(y(x),x),x)-x*(6-x)*diff(y(x),x)+(8-x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{2} \left (\frac {2 \sqrt {1+x}\, c_2 x}{3}+\left (\ln \left (\sqrt {1+x}-1\right )-\ln \left (1+\sqrt {1+x}\right )+\frac {8 \sqrt {1+x}}{3}\right ) c_2 +c_1 \right )}{\left (1+x \right )^{{5}/{2}}} \]

✓ Mathematica. Time used: 0.244 (sec). Leaf size: 109

ode=2*x^2*(1+x)*D[y[x],{x,2}]-x*(6-x)*D[y[x],x]+(8-x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {2-K[1]}{4 K[1]^2+4 K[1]}dK[1]-\frac {1}{2} \int _1^x\left (\frac {7}{2 (K[2]+1)}-\frac {3}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {2-K[1]}{4 K[1]^2+4 K[1]}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x + 1)*Derivative(y(x), (x, 2)) - x*(6 - x)*Derivative(y(x), x) + (8 - x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False