59.1.142 problem 144

Internal problem ID [9314]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 144
Date solved : Monday, January 27, 2025 at 06:01:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 25

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

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

✓ Solution by Mathematica

Time used: 0.210 (sec). Leaf size: 112

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

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