59.1.8 problem 8

Internal problem ID [9180]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 8
Date solved : Monday, January 27, 2025 at 05:51:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 14

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

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

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 48

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

\[ y(x)\to \frac {1}{2} i (2 c_1 x-c_2 x \log (1-i x)+c_2 x \log (1+i x)+2 i c_2) \]