59.1.404 problem 416

Internal problem ID [9576]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 416
Date solved : Monday, January 27, 2025 at 06:04:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }-3 y x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 58

dsolve(diff(y(x),x$2)-x^2*diff(y(x),x)-3*x*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {9 \,{\mathrm e}^{\frac {x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right ) c_{2} x^{3}+9 c_{1} x^{2} {\mathrm e}^{\frac {x^{3}}{3}}+5 \,3^{{2}/{3}} c_{2} \left (x^{3}\right )^{{1}/{3}} \left (x^{3}+2\right )}{9 x} \]

✓ Solution by Mathematica

Time used: 0.063 (sec). Leaf size: 51

DSolve[D[y[x],{x,2}]-x^2*D[y[x],x]-3*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{9} e^{\frac {x^3}{3}} \left (9 c_1 x-3^{2/3} c_2 \sqrt [3]{x^3} \Gamma \left (-\frac {1}{3},\frac {x^3}{3}\right )\right ) \]