Internal problem ID [8028]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 448.
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 {\left (x^{2}-1\right ) \left (y^{\prime }\right )^{2}-y^{2}+1=0} \end {gather*}
✗ Solution by Maple
dsolve((x^2-1)*diff(y(x),x)^2-y(x)^2+1 = 0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 4.701 (sec). Leaf size: 218
DSolve[1 - y[x]^2 + (-1 + x^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{-c_1} \sqrt {e^{2 c_1} \left (\left (2 x^2-1\right ) \cosh (2 c_1)+2 x \sqrt {x^2-1} \sinh (2 c_1)+1\right )}}{\sqrt {2}} \\ y(x)\to \frac {e^{-c_1} \sqrt {e^{2 c_1} \left (\left (2 x^2-1\right ) \cosh (2 c_1)+2 x \sqrt {x^2-1} \sinh (2 c_1)+1\right )}}{\sqrt {2}} \\ y(x)\to -\frac {1}{2} \sqrt {\left (4 x^2-2\right ) \cosh (2 c_1)-4 x \sqrt {x^2-1} \sinh (2 c_1)+2} \\ y(x)\to \frac {1}{2} \sqrt {\left (4 x^2-2\right ) \cosh (2 c_1)-4 x \sqrt {x^2-1} \sinh (2 c_1)+2} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}