59.1.319 problem 326

Internal problem ID [9491]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 326
Date solved : Sunday, March 30, 2025 at 02:36:06 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 36

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

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

✓ Mathematica. Time used: 0.205 (sec). Leaf size: 74

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

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

✓ Sympy. Time used: 0.163 (sec). Leaf size: 31

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

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