59.1.485 problem 501

Internal problem ID [9657]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 501
Date solved : Monday, January 27, 2025 at 06:05:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x^{6} y^{\prime }+7 x^{5} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 62

dsolve(diff(y(x),x$2)+x^6*diff(y(x),x)+7*x^5*y(x)=0,y(x), singsol=all)
 

\[ y = -\frac {\left (-c_{1} {\mathrm e}^{-\frac {x^{7}}{7}} x -c_{2} 7^{{1}/{7}}\right ) \left (-x^{7}\right )^{{6}/{7}}+x^{7} c_{2} {\mathrm e}^{-\frac {x^{7}}{7}} \left (\Gamma \left (\frac {6}{7}\right )-\Gamma \left (\frac {6}{7}, -\frac {x^{7}}{7}\right )\right )}{\left (-x^{7}\right )^{{6}/{7}}} \]

✓ Solution by Mathematica

Time used: 0.080 (sec). Leaf size: 53

DSolve[D[y[x],{x,2}]+x^6*D[y[x],x]+7*x^5*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{49} e^{-\frac {x^7}{7}} \left (49 c_1 x-7^{6/7} c_2 \sqrt [7]{-x^7} \Gamma \left (-\frac {1}{7},-\frac {x^7}{7}\right )\right ) \]