59.1.133 problem 135

Internal problem ID [9305]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 135
Date solved : Wednesday, March 05, 2025 at 07:47:14 AM
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: 0.019 (sec). Leaf size: 26

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);
 

\[ y = \frac {x^{3} \left (2 c_{2} x -c_{2} \ln \left (x \right )+c_{1} \right )}{\left (-1+2 x \right )^{2}} \]

✓ Mathematica. Time used: 0.212 (sec). Leaf size: 95

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]
 

\[ y(x)\to \exp \left (\int _1^x\frac {1}{2 K[1]-4 K[1]^2}dK[1]-\frac {1}{2} \int _1^x\left (\frac {6}{2 K[2]-1}-\frac {5}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {1}{2 K[1]-4 K[1]^2}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

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

False