59.1.226 problem 229

Internal problem ID [9398]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 229
Date solved : Wednesday, March 05, 2025 at 07:48:37 AM
CAS classification : [_Lienard]

\begin{align*} 4 y^{\prime \prime }+x y^{\prime }+4 y&=0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 34

ode:=4*diff(diff(y(x),x),x)+x*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (-12 \operatorname {hypergeom}\left (\left [-\frac {3}{2}\right ], \left [\frac {1}{2}\right ], \frac {x^{2}}{8}\right ) c_{2} +x c_{1} \left (x^{2}-12\right )\right ) {\mathrm e}^{-\frac {x^{2}}{8}}}{12} \]

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 122

ode=4*D[y[x],{x,2}]+x*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{-\frac {x^2}{8}} \left (\sqrt {2 \pi } c_2 \left (x^2-12\right ) x^2 \text {erfi}\left (\frac {\sqrt {x^2}}{2 \sqrt {2}}\right )+4 \sqrt {x^2} \left (2 \sqrt {2} c_1 x^3-c_2 e^{\frac {x^2}{8}} x^2+8 c_2 e^{\frac {x^2}{8}}-24 \sqrt {2} c_1 x\right )\right )}{32 \sqrt {x^2}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) + 4*y(x) + 4*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False