59.1.652 problem 669

Internal problem ID [9824]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 669
Date solved : Monday, January 27, 2025 at 06:14:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 47

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

\[ y = c_{1} \sqrt {2}\, x \left (x^{2}+5\right ) {\mathrm e}^{-\frac {x^{2}}{2}}+\left (x^{4}+6 x^{2}+3\right ) \left (\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right ) c_{1} +c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 43

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

\[ y(x)\to c_1 e^{-\frac {x^2}{2}} \operatorname {HermiteH}\left (-5,\frac {x}{\sqrt {2}}\right )+\frac {1}{3} c_2 \left (x^4+6 x^2+3\right ) \]