59.1.23 problem 23

Internal problem ID [9195]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 23
Date solved : Monday, January 27, 2025 at 05:51:57 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: 0.006 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 0.052 (sec). Leaf size: 69

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

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