##### 4.19.18 $$\left (1-x^2\right ) y'(x)^2=1-y(x)^2$$

ODE
$\left (1-x^2\right ) y'(x)^2=1-y(x)^2$ ODE Classiﬁcation

[_rational, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]

Book solution method
No Missing Variables ODE, Solve for $$y'$$

Mathematica
cpu = 0.269248 (sec), leaf count = 287

$\left \{\left \{y(x)\to -\frac {1}{2} e^{-c_1} \sqrt {2 x^2-2 \sqrt {x^2-1} x+e^{4 c_1} \left (2 x^2+2 \sqrt {x^2-1} x-1\right )-1+2 e^{2 c_1}}\right \},\left \{y(x)\to \frac {1}{2} e^{-c_1} \sqrt {2 x^2-2 \sqrt {x^2-1} x+e^{4 c_1} \left (2 x^2+2 \sqrt {x^2-1} x-1\right )-1+2 e^{2 c_1}}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {e^{-2 c_1} \left (2 x^2+2 \sqrt {x^2-1} x+e^{4 c_1} \left (2 x^2-2 \sqrt {x^2-1} x-1\right )-1+2 e^{2 c_1}\right )}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {e^{-2 c_1} \left (2 x^2+2 \sqrt {x^2-1} x+e^{4 c_1} \left (2 x^2-2 \sqrt {x^2-1} x-1\right )-1+2 e^{2 c_1}\right )}\right \}\right \}$

Maple
cpu = 0. (sec), leaf count = 0 , exception

time expired

Mathematica raw input

DSolve[(1 - x^2)*y'[x]^2 == 1 - y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -1/2*Sqrt[-1 + 2*E^(2*C[1]) + 2*x^2 - 2*x*Sqrt[-1 + x^2] + E^(4*C[1])*
(-1 + 2*x^2 + 2*x*Sqrt[-1 + x^2])]/E^C[1]}, {y[x] -> Sqrt[-1 + 2*E^(2*C[1]) + 2*
x^2 - 2*x*Sqrt[-1 + x^2] + E^(4*C[1])*(-1 + 2*x^2 + 2*x*Sqrt[-1 + x^2])]/(2*E^C[
1])}, {y[x] -> -1/2*Sqrt[(-1 + 2*E^(2*C[1]) + 2*x^2 + 2*x*Sqrt[-1 + x^2] + E^(4*
C[1])*(-1 + 2*x^2 - 2*x*Sqrt[-1 + x^2]))/E^(2*C[1])]}, {y[x] -> Sqrt[(-1 + 2*E^(
2*C[1]) + 2*x^2 + 2*x*Sqrt[-1 + x^2] + E^(4*C[1])*(-1 + 2*x^2 - 2*x*Sqrt[-1 + x^
2]))/E^(2*C[1])]/2}}

Maple raw input

dsolve((-x^2+1)*diff(y(x),x)^2 = 1-y(x)^2, y(x))

Maple raw output

\verbtime expired||