Internal problem ID [5063]
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: 55.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 20
dsolve((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x +c_{2} \sqrt {x -1}\, \sqrt {x +1} \]
✓ Solution by Mathematica
Time used: 0.05 (sec). Leaf size: 87
DSolve[(1-x^2)*y''[x]-x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cosh \left (\frac {\sqrt {1-x^2} \text {ArcTan}\left (\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {x^2-1}}\right )+i c_2 \sinh \left (\frac {\sqrt {1-x^2} \text {ArcTan}\left (\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {x^2-1}}\right ) \\ \end{align*}