59.1.514 problem 530

Internal problem ID [9686]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 530
Date solved : Sunday, March 30, 2025 at 02:44:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 29

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

\[ y = \frac {c_1 \sin \left (\sqrt {x}\, \sqrt {2}\right )+c_2 \cos \left (\sqrt {x}\, \sqrt {2}\right )}{x} \]

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 60

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

\[ y(x)\to \frac {2 c_1 e^{i \sqrt {2} \sqrt {x}}+i \sqrt {2} c_2 e^{-i \sqrt {2} \sqrt {x}}}{2 x} \]

✓ Sympy. Time used: 0.242 (sec). Leaf size: 37

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

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