59.1.50 problem 52

Internal problem ID [9222]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 52
Date solved : Monday, January 27, 2025 at 05:52:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }+6 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 24

dsolve((1+x^2)*diff(y(x),x$2)+6*x*diff(y(x),x)+6*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{2} x^{2}+c_{1} x -c_{2}}{\left (x^{2}+1\right )^{2}} \]

✓ Solution by Mathematica

Time used: 0.346 (sec). Leaf size: 79

DSolve[(1+x^2)*D[y[x],{x,2}]+6*x*D[y[x],x]+6*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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