59.1.722 problem 739

Internal problem ID [9894]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 739
Date solved : Monday, January 27, 2025 at 06:15:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.050 (sec). Leaf size: 29

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

\[ y(x)\to c_2 x \text {arcsinh}(x)-c_2 \sqrt {x^2+1}+c_1 x \]