59.1.564 problem 580

Internal problem ID [9736]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 580
Date solved : Monday, January 27, 2025 at 06:13:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.148 (sec). Leaf size: 96

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

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