59.1.376 problem 383

Internal problem ID [9548]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 383
Date solved : Wednesday, March 05, 2025 at 07:50:46 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 2 x^{2} y^{\prime \prime }+3 x y^{\prime }-x y&=0 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 29

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

\[ y = \frac {c_{1} \sinh \left (\sqrt {x}\, \sqrt {2}\right )+c_{2} \cosh \left (\sqrt {x}\, \sqrt {2}\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.077 (sec). Leaf size: 56

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

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

✓ Sympy. Time used: 0.192 (sec). Leaf size: 41

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

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