59.1.691 problem 708

Internal problem ID [9863]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 708
Date solved : Sunday, March 30, 2025 at 02:48:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.133 (sec). Leaf size: 35

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

\[ y = \frac {c_1 \left (5 x +2\right )+c_2 \,{\mathrm e}^{-\frac {1}{x}} \operatorname {hypergeom}\left (\left [2\right ], \left [-\frac {1}{2}\right ], \frac {1}{x}\right ) x^{{5}/{2}}}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.393 (sec). Leaf size: 65

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

\[ y(x)\to \frac {\sqrt {e} (5 x+2) \left (c_2 \int _1^x\frac {25 e^{-\frac {5}{2}-\frac {1}{K[1]}} K[1]^{5/2}}{(5 K[1]+2)^2}dK[1]+c_1\right )}{5 \sqrt {x}} \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (2*x**3*Derivative(y(x), (x, 2)) + 2*x*y(x) - y(x))/(x*(3*x + 2)) cannot be solved by the factorable group method