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 = 0.0189176 (sec), leaf count = 29
\[\left \{\left \{y(x)\to c_1 x+\sqrt {1-c_1^2}-c_1 \cos ^{-1}\left (c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.084 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) -{\frac {1}{2}\sqrt {2-2\,\cos \left ( 2\,x \right ) }}=0,y \left ( x \right ) ={\it \_C1}\,x-{\it \_C1}\,\arccos \left ( {\it \_C1} \right ) +\sqrt {-{{\it \_C1}}^{2}+1} \right \} \] 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),'implicit')
Maple raw output
y(x)-1/2*(2-2*cos(2*x))^(1/2) = 0, y(x) = _C1*x-_C1*arccos(_C1)+(-_C1^2+1)^(1/2)