59.1.67 problem 69

Internal problem ID [9239]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 69
Date solved : Monday, January 27, 2025 at 05:59:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x^{5} y^{\prime }+6 x^{4} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 63

dsolve(diff(y(x),x$2)+x^5*diff(y(x),x)+6*x^4*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.233 (sec). Leaf size: 53

DSolve[D[y[x],{x,2}]+x^5*D[y[x],x]+6*x^4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{36} e^{-\frac {x^6}{6}} \left (36 c_1 x-6^{5/6} c_2 \sqrt [6]{-x^6} \Gamma \left (-\frac {1}{6},-\frac {x^6}{6}\right )\right ) \]