59.1.408 problem 420

Internal problem ID [9580]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 420
Date solved : Monday, January 27, 2025 at 06:04:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+6 x \right ) y^{\prime \prime }+\left (3 x +9\right ) y^{\prime }-3 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 30

dsolve((x^2+6*x)*diff(y(x),x$2)+(3*x+9)*diff(y(x),x)-3*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \left (x +3\right )+\frac {c_{2} \left (2 x^{2}+12 x +9\right )}{\sqrt {x}\, \sqrt {6+x}} \]

✓ Solution by Mathematica

Time used: 0.673 (sec). Leaf size: 103

DSolve[(x^2+6*x)*D[y[x],{x,2}]+(3*x+9)*D[y[x],x]-3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\left (9 \sqrt {\pi } c_2 \sqrt [4]{-x (x+6)} Q_{\frac {3}{2}}^{\frac {1}{2}}\left (\frac {x}{3}+1\right )+\sqrt {6} c_1 \left (2 x^2+12 x+9\right )\right ) \exp \left (\int _1^x-\frac {K[1]+3}{2 K[1] (K[1]+6)}dK[1]\right )}{9 \sqrt {\pi } \sqrt [4]{-x (x+6)}} \]