59.1.750 problem 772

Internal problem ID [9922]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 772
Date solved : Sunday, March 30, 2025 at 02:49:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.227 (sec). Leaf size: 73

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

\[ y = \frac {c_1 \sqrt {\frac {\left (2 \sqrt {x}+i\right ) \left (4 x +1\right )}{-2 \sqrt {x}+i}}\, {\mathrm e}^{2 i \sqrt {x}}+c_2 \sqrt {\frac {\left (-2 \sqrt {x}+i\right ) \left (4 x +1\right )}{2 \sqrt {x}+i}}\, {\mathrm e}^{-2 i \sqrt {x}}}{x} \]

✓ Mathematica. Time used: 0.213 (sec). Leaf size: 77

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

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

✓ Sympy. Time used: 0.216 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = \frac {C_{1} J_{\frac {3}{2}}\left (2 \sqrt {x}\right ) + C_{2} Y_{\frac {3}{2}}\left (2 \sqrt {x}\right )}{\sqrt [4]{x}} \]