Internal problem ID [8783]
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)]`]]
\[ \boxed {\left (x^{2}-1\right ) {y^{\prime }}^{2}-y^{2}=-1} \]
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 162
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 {y \left (x \right )^{2}-1}\, \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}}-\frac {\int _{}^{x}\frac {\sqrt {\left (\textit {\_a}^{2}-1\right ) \left (y \left (x \right )^{2}-1\right )}}{\textit {\_a}^{2}-1}d \textit {\_a}}{\sqrt {y \left (x \right )-1}\, \sqrt {y \left (x \right )+1}}+c_{1} &= 0 \\ \frac {\sqrt {y \left (x \right )^{2}-1}\, \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}}+\frac {\int _{}^{x}\frac {\sqrt {\left (\textit {\_a}^{2}-1\right ) \left (y \left (x \right )^{2}-1\right )}}{\textit {\_a}^{2}-1}d \textit {\_a}}{\sqrt {y \left (x \right )-1}\, \sqrt {y \left (x \right )+1}}+c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.099 (sec). Leaf size: 297
DSolve[1 - y[x]^2 + (-1 + x^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} e^{-c_1} \sqrt {2 x^2-2 \sqrt {x^2-1} x+e^{4 c_1} \left (2 x^2+2 \sqrt {x^2-1} x-1\right )-1+2 e^{2 c_1}} \\ y(x)\to \frac {1}{2} e^{-c_1} \sqrt {2 x^2-2 \sqrt {x^2-1} x+e^{4 c_1} \left (2 x^2+2 \sqrt {x^2-1} x-1\right )-1+2 e^{2 c_1}} \\ y(x)\to -\frac {1}{2} \sqrt {e^{-2 c_1} \left (2 x^2+2 \sqrt {x^2-1} x+e^{4 c_1} \left (2 x^2-2 \sqrt {x^2-1} x-1\right )-1+2 e^{2 c_1}\right )} \\ y(x)\to \frac {1}{2} \sqrt {e^{-2 c_1} \left (2 x^2+2 \sqrt {x^2-1} x+e^{4 c_1} \left (2 x^2-2 \sqrt {x^2-1} x-1\right )-1+2 e^{2 c_1}\right )} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}