59.1.649 problem 666

Internal problem ID [9821]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 666
Date solved : Monday, January 27, 2025 at 06:14:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x y^{\prime }+\left (x +2\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 78

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

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

✓ Solution by Mathematica

Time used: 0.248 (sec). Leaf size: 63

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

\[ y(x)\to e^{x-\frac {x^2}{2}} \left (x^2-4 x+3\right ) \left (c_2 \int _1^x\frac {e^{\frac {1}{2} (K[1]-4) K[1]}}{(K[1]-3)^2 (K[1]-1)^2}dK[1]+c_1\right ) \]