59.1.9 problem 9

Internal problem ID [9181]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 9
Date solved : Monday, January 27, 2025 at 05:51:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 35

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

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