Internal problem ID [3642]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 914.
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 (1-x^{2}\right ) \left (y^{\prime }\right )^{2}-1+y^{2}=0} \end {gather*}
✗ Solution by Maple
dsolve((-x^2+1)*diff(y(x),x)^2 = 1-y(x)^2,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 4.805 (sec). Leaf size: 218
DSolve[(1-x^2) (y'[x])^2==1-y[x]^2,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*}