59.1.510 problem 526

Internal problem ID [9682]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 526
Date solved : Sunday, March 30, 2025 at 02:44:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 25

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

\[ y = \sqrt {x}\, \left (c_1 \sinh \left (2 \sqrt {x}\right )+c_2 \cosh \left (2 \sqrt {x}\right )\right ) \]

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 41

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

\[ y(x)\to \frac {1}{2} e^{-2 \sqrt {x}} \sqrt {x} \left (2 c_1 e^{4 \sqrt {x}}-c_2\right ) \]

✓ Sympy. Time used: 0.218 (sec). Leaf size: 34

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

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