59.1.226 problem 229

Internal problem ID [9398]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 229
Date solved : Monday, January 27, 2025 at 06:02:19 PM
CAS classification : [_Lienard]

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

✓ Solution by Maple

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

dsolve(4*diff(y(x),x$2)+x*diff(y(x),x)+4*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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