59.1.546 problem 562

Internal problem ID [9718]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 562
Date solved : Monday, January 27, 2025 at 06:13:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.248 (sec). Leaf size: 147

dsolve(x^2*(1+x+x^2)*diff(y(x),x$2)-x*(1-4*x-2*x^2)*diff(y(x),x)+y(x)=0,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-\frac {7 \sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}} x \left (c_{2} \left (-2 x +i \sqrt {3}-1\right )^{-\frac {1}{4}-\frac {7 i \sqrt {3}}{12}} \left (2 x +i \sqrt {3}+1\right )^{\frac {3}{4}+\frac {7 i \sqrt {3}}{12}} \operatorname {hypergeom}\left (\left [1, \frac {1}{2}+\frac {7 i \sqrt {3}}{6}\right ], \left [\frac {3}{2}+\frac {7 i \sqrt {3}}{6}\right ], \frac {-i \sqrt {3}\, x +x +2}{i \sqrt {3}\, x +x +2}\right )+c_{1} \left (-2 x +i \sqrt {3}-1\right )^{\frac {1}{4}+\frac {7 i \sqrt {3}}{12}} \left (2 x +i \sqrt {3}+1\right )^{\frac {1}{4}-\frac {7 i \sqrt {3}}{12}}\right )}{\left (x^{2}+x +1\right )^{{3}/{4}}} \]

✓ Solution by Mathematica

Time used: 0.552 (sec). Leaf size: 130

DSolve[x^2*(1+x+x^2)*D[y[x],{x,2}]-x*(1-4*x-2*x^2)*D[y[x],x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {1}{2} \left (\frac {K[1]-3}{K[1]^2+K[1]+1}+\frac {1}{K[1]}\right )dK[1]-\frac {1}{2} \int _1^x\left (\frac {3 K[2]+5}{K[2]^2+K[2]+1}-\frac {1}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {2 K[1]^2-2 K[1]+1}{2 K[1] \left (K[1]^2+K[1]+1\right )}dK[1]\right )dK[3]+c_1\right ) \]