4.15.40 \(y'(x) \cos (y(x)) (\cos (y(x))-\sin (A) \sin (x))+\cos (x) (\cos (x)-\sin (A) \sin (y(x)))=0\)

ODE
\[ y'(x) \cos (y(x)) (\cos (y(x))-\sin (A) \sin (x))+\cos (x) (\cos (x)-\sin (A) \sin (y(x)))=0 \] ODE Classification

[_exact]

Book solution method
Exact equation

Mathematica
cpu = 0.143213 (sec), leaf count = 32

\[\text {Solve}\left [c_1+2 y(x)+\sin (2 y(x))+2 x+\sin (2 x)=4 \sin (A) \sin (x) \sin (y(x)),y(x)\right ]\]

Maple
cpu = 0.271 (sec), leaf count = 33

\[ \left \{ {\frac { \left (-2\,\sin \relax (A ) \sin \relax (x ) +\cos \left (y \relax (x ) \right ) \right ) \sin \left (y \relax (x ) \right ) }{2}}+{\frac {\cos \relax (x ) \sin \relax (x ) }{2}}+{\frac {x}{2}}+{\it \_C1}+{\frac {y \relax (x ) }{2}}=0 \right \} \] Mathematica raw input

DSolve[Cos[x]*(Cos[x] - Sin[A]*Sin[y[x]]) + Cos[y[x]]*(Cos[y[x]] - Sin[A]*Sin[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[2*x + C[1] + Sin[2*x] + Sin[2*y[x]] + 2*y[x] == 4*Sin[A]*Sin[x]*Sin[y[x]],
 y[x]]

Maple raw input

dsolve(diff(y(x),x)*cos(y(x))*(cos(y(x))-sin(A)*sin(x))+cos(x)*(cos(x)-sin(A)*sin(y(x))) = 0, y(x),'implicit')

Maple raw output

1/2*(-2*sin(A)*sin(x)+cos(y(x)))*sin(y(x))+1/2*cos(x)*sin(x)+1/2*x+_C1+1/2*y(x) 
= 0