59.1.104 problem 106

Internal problem ID [9276]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 106
Date solved : Monday, January 27, 2025 at 06:00:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.085 (sec). Leaf size: 35

dsolve(2*x^2*(2+x^2)*diff(y(x),x$2)-x*(12-7*x^2)*diff(y(x),x)+(7+3*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{1} x^{{7}/{2}}}{\left (2 x^{2}+4\right )^{{9}/{4}}}+c_{2} \sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1\right ], \left [-\frac {1}{2}\right ], -\frac {x^{2}}{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.316 (sec). Leaf size: 117

DSolve[2*x^2*(2+x^2)*D[y[x],{x,2}]-x*(12-7*x^2)*D[y[x],x]+(7+3*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {3 K[1]^2+16}{4 K[1]^3+8 K[1]}dK[1]-\frac {1}{2} \int _1^x\left (\frac {13 K[2]}{2 \left (K[2]^2+2\right )}-\frac {3}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {3 K[1]^2+16}{4 K[1]^3+8 K[1]}dK[1]\right )dK[3]+c_1\right ) \]