59.1.332 problem 339

Internal problem ID [9504]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 339
Date solved : Monday, January 27, 2025 at 06:03:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 56

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

\[ y = -\frac {\left (-c_{2} 3^{{1}/{3}} {\mathrm e}^{\frac {x^{3}}{3}}-c_{1} x \right ) \left (-x^{3}\right )^{{2}/{3}}+c_{2} x^{3} \left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )\right )}{\left (-x^{3}\right )^{{2}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.055 (sec). Leaf size: 41

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

\[ y(x)\to c_1 x-\frac {c_2 \sqrt [3]{-x^3} \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )}{3 \sqrt [3]{3}} \]