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 = 0.0534169 (sec), leaf count = 69
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \sqrt {1-\text {$\#$1}^2}+\sin ^{-1}(\text {$\#$1})\right )\& \right ]\left [c_1-x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \sqrt {1-\text {$\#$1}^2}+\sin ^{-1}(\text {$\#$1})\right )\& \right ]\left [c_1+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.449 (sec), leaf count = 53
\[ \left \{ x-{\frac {y \left ( x \right ) }{2}\sqrt {1- \left ( y \left ( x \right ) \right ) ^{2}}}-{\frac {\arcsin \left ( y \left ( x \right ) \right ) }{2}}-{\it \_C1}=0,x+{\frac {y \left ( x \right ) }{2}\sqrt {1- \left ( y \left ( x \right ) \right ) ^{2}}}+{\frac {\arcsin \left ( y \left ( x \right ) \right ) }{2}}-{\it \_C1}=0 \right \} \] 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),'implicit')
Maple raw output
x-1/2*y(x)*(1-y(x)^2)^(1/2)-1/2*arcsin(y(x))-_C1 = 0, x+1/2*y(x)*(1-y(x)^2)^(1/2)+1/2*arcsin(y(x))-_C1 = 0