Internal problem ID [8036]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 456.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {x \left (x^{2}-1\right ) \left (y^{\prime }\right )^{2}+2 \left (-x^{2}+1\right ) y y^{\prime }+x y^{2}-x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 33
dsolve(x*(x^2-1)*diff(y(x),x)^2+2*(-x^2+1)*y(x)*diff(y(x),x)+x*y(x)^2-x = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -x \\ y \relax (x ) = x \\ y \relax (x ) = \sqrt {-c_{1}^{2}+1}+\sqrt {x^{2}-1}\, c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.983 (sec). Leaf size: 73
DSolve[-x + x*y[x]^2 + 2*(1 - x^2)*y[x]*y'[x] + x*(-1 + x^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \cos \left (2 \text {ArcTan}\left (\sqrt {\frac {x-1}{x+1}}\right )+i c_1\right ) \\ y(x)\to x \cos \left (2 \text {ArcTan}\left (\sqrt {\frac {x-1}{x+1}}\right )-i c_1\right ) \\ y(x)\to -x \\ y(x)\to x \\ \end{align*}