59.1.137 problem 139

Internal problem ID [9309]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 139
Date solved : Sunday, March 30, 2025 at 02:32:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5+4 x \right ) y^{\prime }+\left (9+4 x \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 54

ode:=x^2*(-2*x+1)*diff(diff(y(x),x),x)-x*(5+4*x)*diff(y(x),x)+(9+4*x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{3} \left (-6 c_2 \left (\frac {1}{8}+x \right ) \ln \left (x \right )+c_2 \,x^{4}-4 c_2 \,x^{3}+9 c_2 \,x^{2}+\left (8 c_1 +\frac {609 c_2}{512}\right ) x +c_1 -\frac {9375 c_2}{4096}\right )}{\left (-1+2 x \right )^{6}} \]

✓ Mathematica. Time used: 0.489 (sec). Leaf size: 129

ode=x^2*(1-2*x)*D[y[x],{x,2}]-x*(5+4*x)*D[y[x],x]+(9+4*x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{8} (8 x+1) \exp \left (\int _1^x\frac {8 K[1]+1}{2 K[1]-4 K[1]^2}dK[1]-\frac {1}{2} \int _1^x-\frac {4 K[2]+5}{K[2]-2 K[2]^2}dK[2]\right ) \left (c_2 \int _1^x\frac {64 \exp \left (-2 \int _1^{K[3]}\frac {8 K[1]+1}{2 K[1]-4 K[1]^2}dK[1]\right )}{(8 K[3]+1)^2}dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - 2*x)*Derivative(y(x), (x, 2)) - x*(4*x + 5)*Derivative(y(x), x) + (4*x + 9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False