ODE
\[ \left (1-y(x)^2\right ) y'(x)^2=1 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.246075 (sec), leaf count = 69
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \sqrt {1-\text {$\#$1}^2}+\sin ^{-1}(\text {$\#$1})\right )\& \right ][-x+c_1]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \sqrt {1-\text {$\#$1}^2}+\sin ^{-1}(\text {$\#$1})\right )\& \right ][x+c_1]\right \}\right \}\]
Maple ✓
cpu = 1.178 (sec), leaf count = 48
\[\left [y \left (x \right ) = \sin \left (\RootOf \left (\sin \left (\textit {\_Z} \right ) \sqrt {\frac {\cos \left (2 \textit {\_Z} \right )}{2}+\frac {1}{2}}+\textit {\_Z} +2 \textit {\_C1} -2 x \right )\right ), y \left (x \right ) = \sin \left (\RootOf \left (-\sin \left (\textit {\_Z} \right ) \sqrt {\frac {\cos \left (2 \textit {\_Z} \right )}{2}+\frac {1}{2}}-\textit {\_Z} +2 \textit {\_C1} -2 x \right )\right )\right ]\] Mathematica raw input
DSolve[(1 - y[x]^2)*y'[x]^2 == 1,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(ArcSin[#1] + #1*Sqrt[1 - #1^2])/2 & ][-x + C[1]]}, {y[x] -> InverseFunction[(ArcSin[#1] + #1*Sqrt[1 - #1^2])/2 & ][x + C[1]]}}
Maple raw input
dsolve((1-y(x)^2)*diff(y(x),x)^2 = 1, y(x))
Maple raw output
[y(x) = sin(RootOf(sin(_Z)*(cos(_Z)^2)^(1/2)+_Z+2*_C1-2*x)), y(x) = sin(RootOf(-sin(_Z)*(cos(_Z)^2)^(1/2)-_Z+2*_C1-2*x))]