4.19.18 $(1-x^2)y^{\prime }(x{)}^2=1-y(x{)}^2$

ODE
$$(1-x^2)y^{\prime }(x{)}^2=1-y(x{)}^2$$ ODE Classification

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

Book solution method
No Missing Variables ODE, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.0818556 (sec), leaf count = 88

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

Maple ✓
cpu = 251.375 (sec), leaf count = 166

$$\{{\left(y\left(x\right)\right)}^2-1=0,1\sqrt{(y\left(x\right)-1)(1+y\left(x\right))}\ln (y\left(x\right)+\sqrt{{\left(y\left(x\right)\right)}^2-1})\frac{1}{\sqrt{y\left(x\right)-1}}\frac{1}{\sqrt{1+y\left(x\right)}}+{\int }^x\frac{1}{{\mathit{\_}\mathit{a}}^2-1}\sqrt{({\mathit{\_}\mathit{a}}^2-1)({\left(y\left(x\right)\right)}^2-1)}\frac{1}{\sqrt{y\left(x\right)-1}}\frac{1}{\sqrt{1+y\left(x\right)}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}=0,1\sqrt{(y\left(x\right)-1)(1+y\left(x\right))}\ln (y\left(x\right)+\sqrt{{\left(y\left(x\right)\right)}^2-1})\frac{1}{\sqrt{y\left(x\right)-1}}\frac{1}{\sqrt{1+y\left(x\right)}}+{\int }^x-\frac{1}{{\mathit{\_}\mathit{a}}^2-1}\sqrt{({\mathit{\_}\mathit{a}}^2-1)({\left(y\left(x\right)\right)}^2-1)}\frac{1}{\sqrt{y\left(x\right)-1}}\frac{1}{\sqrt{1+y\left(x\right)}}d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

y(x)^2-1 = 0, ((y(x)-1)*(1+y(x)))^(1/2)/(y(x)-1)^(1/2)/(1+y(x))^(1/2)*ln(y(x)+(y
(x)^2-1)^(1/2))+Intat(-1/(_a^2-1)*((_a^2-1)*(y(x)^2-1))^(1/2)/(y(x)-1)^(1/2)/(1+
y(x))^(1/2),_a = x)+_C1 = 0, ((y(x)-1)*(1+y(x)))^(1/2)/(y(x)-1)^(1/2)/(1+y(x))^(
1/2)*ln(y(x)+(y(x)^2-1)^(1/2))+Intat(1/(_a^2-1)*((_a^2-1)*(y(x)^2-1))^(1/2)/(y(x
)-1)^(1/2)/(1+y(x))^(1/2),_a = x)+_C1 = 0