ODE
\[ \left (y'(x)^2+1\right ) \sin ^2\left (y(x)-x y'(x)\right )=1 \] ODE Classification
[_Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 0.0377712 (sec), leaf count = 59
\[\left \{\left \{y(x)\to c_1 x-\frac {1}{2} \cos ^{-1}\left (\frac {c_1^2-1}{c_1^2+1}\right )\right \},\left \{y(x)\to c_1 x+\frac {1}{2} \cos ^{-1}\left (\frac {c_1^2-1}{c_1^2+1}\right )\right \}\right \}\]
Maple ✓
cpu = 0.862 (sec), leaf count = 151
\[ \left \{ y \left ( x \right ) -\sqrt {1-x}\sqrt {{x}^{-1}}x-\arcsin \left ( \sqrt {{x}^{-1}}x \right ) =0,y \left ( x \right ) +\sqrt {1-x}\sqrt {{x}^{-1}}x+\arcsin \left ( \sqrt {{x}^{-1}}x \right ) =0,y \left ( x \right ) -\sqrt {1+x}\sqrt {-{x}^{-1}}x-\arcsin \left ( \sqrt {-{x}^{-1}}x \right ) =0,y \left ( x \right ) +\sqrt {1+x}\sqrt {-{x}^{-1}}x+\arcsin \left ( \sqrt {-{x}^{-1}}x \right ) =0,y \left ( x \right ) ={\it \_C1}\,x-\arcsin \left ( {\frac {1}{\sqrt {{{\it \_C1}}^{2}+1}}} \right ) ,y \left ( x \right ) ={\it \_C1}\,x+\arcsin \left ( {\frac {1}{\sqrt {{{\it \_C1}}^{2}+1}}} \right ) \right \} \] Mathematica raw input
DSolve[Sin[y[x] - x*y'[x]]^2*(1 + y'[x]^2) == 1,y[x],x]
Mathematica raw output
{{y[x] -> -ArcCos[(-1 + C[1]^2)/(1 + C[1]^2)]/2 + x*C[1]}, {y[x] -> ArcCos[(-1 + C[1]^2)/(1 + C[1]^2)]/2 + x*C[1]}}
Maple raw input
dsolve((1+diff(y(x),x)^2)*sin(x*diff(y(x),x)-y(x))^2 = 1, y(x),'implicit')
Maple raw output
y(x)-(1-x)^(1/2)*(1/x)^(1/2)*x-arcsin((1/x)^(1/2)*x) = 0, y(x)+(1-x)^(1/2)*(1/x)^(1/2)*x+arcsin((1/x)^(1/2)*x) = 0, y(x)-(1+x)^(1/2)*(-1/x)^(1/2)*x-arcsin((-1/x)^(1/2)*x) = 0, y(x)+(1+x)^(1/2)*(-1/x)^(1/2)*x+arcsin((-1/x)^(1/2)*x) = 0, y(x) = _C1*x-arcsin(1/(_C1^2+1)^(1/2)), y(x) = _C1*x+arcsin(1/(_C1^2+1)^(1/2))