59.1.520 problem 536

Internal problem ID [9692]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 536
Date solved : Monday, January 27, 2025 at 06:13:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 26

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

\[ y = \frac {x \left (c_{1} x^{3}+3 c_{2} x^{2}+c_{2} \right )}{\left (x^{2}+1\right )^{2}} \]

✓ Solution by Mathematica

Time used: 0.197 (sec). Leaf size: 101

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

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