Internal problem ID [8026]
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)]`]]
\[ \boxed {\left (x^{2}-1\right ) {y^{\prime }}^{2}-y^{2}+1=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 166
dsolve((x^2-1)*diff(y(x),x)^2-y(x)^2+1 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -1 \\ y \left (x \right ) = 1 \\ \frac {\sqrt {\left (y \left (x \right )-1\right ) \left (y \left (x \right )+1\right )}\, \ln \left (y \left (x \right )+\sqrt {y \left (x \right )^{2}-1}\right )}{\sqrt {y \left (x \right )-1}\, \sqrt {y \left (x \right )+1}}+\int _{}^{x}-\frac {\sqrt {\left (\textit {\_a}^{2}-1\right ) \left (y \left (x \right )^{2}-1\right )}}{\left (\textit {\_a}^{2}-1\right ) \sqrt {y \left (x \right )-1}\, \sqrt {y \left (x \right )+1}}d \textit {\_a} +c_{1} = 0 \\ \frac {\sqrt {\left (y \left (x \right )-1\right ) \left (y \left (x \right )+1\right )}\, \ln \left (y \left (x \right )+\sqrt {y \left (x \right )^{2}-1}\right )}{\sqrt {y \left (x \right )-1}\, \sqrt {y \left (x \right )+1}}+\int _{}^{x}\frac {\sqrt {\left (\textit {\_a}^{2}-1\right ) \left (y \left (x \right )^{2}-1\right )}}{\left (\textit {\_a}^{2}-1\right ) \sqrt {y \left (x \right )-1}\, \sqrt {y \left (x \right )+1}}d \textit {\_a} +c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.174 (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*}