59.1.49 problem 51

Internal problem ID [9221]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 51
Date solved : Monday, January 27, 2025 at 05:52:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 53

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

\[ y = \frac {-c_{2} \left (x -2\right ) \sqrt {x^{2}+2}\, \left (x +2\right ) \sqrt {2}+\left (x^{2}+8\right ) x^{2} \left (\operatorname {arctanh}\left (\frac {\sqrt {2}}{\sqrt {x^{2}+2}}\right ) c_{2} +c_{1} \right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.628 (sec). Leaf size: 112

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

\[ y(x)\to \left (x^2+8\right ) \exp \left (\int _1^x\frac {3}{K[1]^3+2 K[1]}dK[1]-\frac {1}{2} \int _1^x\frac {2 K[2]^2+10}{K[2]^3+2 K[2]}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}\frac {3}{K[1]^3+2 K[1]}dK[1]\right )}{\left (K[3]^2+8\right )^2}dK[3]+c_1\right ) \]