59.1.165 problem 167

Internal problem ID [9337]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 167
Date solved : Sunday, March 30, 2025 at 02:32:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-2 x \right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 x \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 43

ode:=x^2*(-2*x+1)*diff(diff(y(x),x),x)+x*(8-9*x)*diff(y(x),x)+(6-3*x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {48 \left (-\frac {1}{2}+x \right )^{4} \left (x +\frac {4}{3}\right ) c_1 \sqrt {-1+2 x}+231 c_2 \left (x^{3}-\frac {6}{7} x^{2}+\frac {2}{7} x -\frac {8}{231}\right )}{x^{6}} \]

✓ Mathematica. Time used: 0.512 (sec). Leaf size: 130

ode=x^2*(1-2*x)*D[y[x],{x,2}]+x*(8-9*x)*D[y[x],x]+(6-3*x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{3} (3 x+4) \exp \left (\int _1^x-\frac {3 K[1]+4}{2 K[1]-4 K[1]^2}dK[1]-\frac {1}{2} \int _1^x\frac {8-9 K[2]}{K[2]-2 K[2]^2}dK[2]\right ) \left (c_2 \int _1^x\frac {9 \exp \left (-2 \int _1^{K[3]}-\frac {3 K[1]+4}{2 K[1]-4 K[1]^2}dK[1]\right )}{(3 K[3]+4)^2}dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - 2*x)*Derivative(y(x), (x, 2)) + x*(8 - 9*x)*Derivative(y(x), x) + (6 - 3*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False