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 = 0.0818556 (sec), leaf count = 88

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

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

\[ \left \{ \left (y \relax (x ) \right ) ^{2}-1=0,{1\sqrt { \left (y \relax (x ) -1 \right ) \left (1+y \relax (x ) \right ) }\ln \left (y \relax (x ) +\sqrt { \left (y \relax (x ) \right ) ^{2}-1} \right ) {\frac {1}{\sqrt {y \relax (x ) -1}}}{\frac {1}{\sqrt {1+y \relax (x ) }}}}+\int ^{x}\!{\frac {1}{{{\it \_a}}^{2}-1}\sqrt { \left ({{\it \_a}}^{2}-1 \right ) \left (\left (y \relax (x ) \right ) ^{2}-1 \right ) }{\frac {1}{\sqrt {y \relax (x ) -1}}}{\frac {1}{\sqrt {1+y \relax (x ) }}}}{d{\it \_a}}+{\it \_C1}=0,{1\sqrt { \left (y \relax (x ) -1 \right ) \left (1+y \relax (x ) \right ) }\ln \left (y \relax (x ) +\sqrt { \left (y \relax (x ) \right ) ^{2}-1} \right ) {\frac {1}{\sqrt {y \relax (x ) -1}}}{\frac {1}{\sqrt {1+y \relax (x ) }}}}+\int ^{x}\!-{\frac {1}{{{\it \_a}}^{2}-1}\sqrt { \left ({{\it \_a}}^{2}-1 \right ) \left (\left (y \relax (x ) \right ) ^{2}-1 \right ) }{\frac {1}{\sqrt {y \relax (x ) -1}}}{\frac {1}{\sqrt {1+y \relax (x ) }}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] 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