Internal problem ID [5068]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 2. Linear homogeneous equations. Section 2.2 problems. page 95
Problem number: 60.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 15
dsolve((1+x^2)*diff(y(x),x$2)+x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sin \left (\arcsinh \relax (x )\right )+c_{2} \cos \left (\arcsinh \relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 18
DSolve[(1+x^2)*y''[x]+x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cos \left (\sinh ^{-1}(x)\right )+c_2 \sin \left (\sinh ^{-1}(x)\right ) \\ \end{align*}