59.1.291 problem 294

Internal problem ID [9463]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 294
Date solved : Monday, January 27, 2025 at 06:03:06 PM
CAS classification : [_Hermite]

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

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 37

dsolve(diff(y(x),x$2)-x*diff(y(x),x)-3*y(x)=0,y(x), singsol=all)
 

\[ y = \left (x^{2}+1\right ) \left (\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right ) c_{1} +c_{2} \right ) {\mathrm e}^{\frac {x^{2}}{2}}+\sqrt {2}\, c_{1} x \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 35

DSolve[D[y[x],{x,2}]-x*D[y[x],x]-3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {HermiteH}\left (-3,\frac {x}{\sqrt {2}}\right )+c_2 e^{\frac {x^2}{2}} \left (x^2+1\right ) \]