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 = 0.258827 (sec), leaf count = 59
\[\left \{\left \{y(x)\to c_1 x-\frac {1}{2} \cos ^{-1}\left (\frac {-1+c_1{}^2}{1+c_1{}^2}\right )\right \},\left \{y(x)\to c_1 x+\frac {1}{2} \cos ^{-1}\left (\frac {-1+c_1{}^2}{1+c_1{}^2}\right )\right \}\right \}\]
Maple ✓
cpu = 0.823 (sec), leaf count = 147
\[\left [y \left (x \right ) = -x \sqrt {1-x}\, \sqrt {\frac {1}{x}}-\arcsin \left (\sqrt {\frac {1}{x}}\, x \right ), y \left (x \right ) = x \sqrt {1-x}\, \sqrt {\frac {1}{x}}+\arcsin \left (\sqrt {\frac {1}{x}}\, x \right ), y \left (x \right ) = -x \sqrt {x +1}\, \sqrt {-\frac {1}{x}}-\arcsin \left (\sqrt {-\frac {1}{x}}\, x \right ), y \left (x \right ) = x \sqrt {x +1}\, \sqrt {-\frac {1}{x}}+\arcsin \left (\sqrt {-\frac {1}{x}}\, x \right ), y \left (x \right ) = \textit {\_C1} x -\arcsin \left (\frac {1}{\sqrt {\textit {\_C1}^{2}+1}}\right ), y \left (x \right ) = \textit {\_C1} x +\arcsin \left (\frac {1}{\sqrt {\textit {\_C1}^{2}+1}}\right )\right ]\] Mathematica raw input
DSolve[Sin[y[x] - x*y'[x]]^2*(1 + y'[x]^2) == 1,y[x],x]
Mathematica raw output
{{y[x] -> -1/2*ArcCos[(-1 + C[1]^2)/(1 + C[1]^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))
Maple raw output
[y(x) = -x*(1-x)^(1/2)*(1/x)^(1/2)-arcsin((1/x)^(1/2)*x), y(x) = x*(1-x)^(1/2)*(1/x)^(1/2)+arcsin((1/x)^(1/2)*x), y(x) = -x*(x+1)^(1/2)*(-1/x)^(1/2)-arcsin((-1/x)^(1/2)*x), y(x) = x*(x+1)^(1/2)*(-1/x)^(1/2)+arcsin((-1/x)^(1/2)*x), y(x) = _C1*x-arcsin(1/(_C1^2+1)^(1/2)), y(x) = _C1*x+arcsin(1/(_C1^2+1)^(1/2))]