ODE
\[ -\sqrt {1-y'(x)^2}+y'(x) \left (\cos ^{-1}\left (y'(x)\right )-x\right )+y(x)=0 \] ODE Classification
[_Clairaut]
Book solution method
Clairaut’s equation and related types, main form
Mathematica ✓
cpu = 0.16051 (sec), leaf count = 29
\[\left \{\left \{y(x)\to c_1 x+\sqrt {1-c_1{}^2}-c_1 \cos ^{-1}(c_1)\right \}\right \}\]
Maple ✓
cpu = 0.14 (sec), leaf count = 35
\[\left [y \left (x \right ) = \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}, y \left (x \right ) = \textit {\_C1} x -\textit {\_C1} \arccos \left (\textit {\_C1} \right )+\sqrt {-\textit {\_C1}^{2}+1}\right ]\] Mathematica raw input
DSolve[y[x] + (-x + ArcCos[y'[x]])*y'[x] - Sqrt[1 - y'[x]^2] == 0,y[x],x]
Mathematica raw output
{{y[x] -> x*C[1] - ArcCos[C[1]]*C[1] + Sqrt[1 - C[1]^2]}}
Maple raw input
dsolve(diff(y(x),x)*(arccos(diff(y(x),x))-x)-(1-diff(y(x),x)^2)^(1/2)+y(x) = 0, y(x))
Maple raw output
[y(x) = (1/2-1/2*cos(2*x))^(1/2), y(x) = _C1*x-_C1*arccos(_C1)+(-_C1^2+1)^(1/2)]