59.1.523 problem 539

Internal problem ID [9695]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 539
Date solved : Monday, January 27, 2025 at 06:13:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.089 (sec). Leaf size: 35

dsolve(3*x^2*(2-x^2)*diff(y(x),x$2)+x*(1-11*x^2)*diff(y(x),x)+(1-5*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{1} \sqrt {x}}{\left (-2 x^{2}+4\right )^{{3}/{4}}}+c_{2} x^{{1}/{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, 1\right ], \left [\frac {11}{12}\right ], \frac {x^{2}}{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.286 (sec). Leaf size: 118

DSolve[3*x^2*(2-x^2)*D[y[x],{x,2}]+x*(1-11*x^2)*D[y[x],x]+(1-5*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {7-5 K[1]^2}{12 K[1]-6 K[1]^3}dK[1]-\frac {1}{2} \int _1^x\frac {1-11 K[2]^2}{6 K[2]-3 K[2]^3}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {7-5 K[1]^2}{12 K[1]-6 K[1]^3}dK[1]\right )dK[3]+c_1\right ) \]