59.1.125 problem 127

Internal problem ID [9297]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 127
Date solved : Monday, January 27, 2025 at 06:01:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (1+x \right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 2.609 (sec). Leaf size: 143

dsolve(4*x^2*(1+x+x^2)*diff(y(x),x$2)+12*x^2*(1+x)*diff(y(x),x)+(1+3*x+3*x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.512 (sec). Leaf size: 120

DSolve[4*x^2*(1+x+x^2)*D[y[x],{x,2}]+12*x^2*(1+x)*D[y[x],x]+(1+3*x+3*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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