59.1.568 problem 584

Internal problem ID [9740]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 584
Date solved : Monday, January 27, 2025 at 06:13:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 23

dsolve(x^2*diff(y(x),x$2)-x*(1-x^2)*diff(y(x),x)+(1+x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 35

DSolve[x^2*D[y[x],{x,2}]-x*(1-x^2)*D[y[x],x]+(1+x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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