59.1.794 problem 817

Internal problem ID [9966]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 817
Date solved : Monday, January 27, 2025 at 06:16:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 52

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

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

✓ Solution by Mathematica

Time used: 0.103 (sec). Leaf size: 78

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

\[ y(x)\to \frac {1}{2} e^{-x} \left (-\sqrt {2 \pi } c_2 \sqrt {(x+2)^2} \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )+2 \sqrt {2} c_1 (x+2)+2 c_2 e^{\frac {1}{2} (x+2)^2}\right ) \]