4.15.40 $y^{\prime }(x)\cos (y(x))(\cos (y(x))-\sin (A)\sin (x))+\cos (x)(\cos (x)-\sin (A)\sin (y(x)))=0$

ODE
$$y^{\prime }(x)\cos (y(x))(\cos (y(x))-\sin (A)\sin (x))+\cos (x)(\cos (x)-\sin (A)\sin (y(x)))=0$$ ODE Classification

[_exact]

Book solution method
Exact equation

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

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

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

$$\{\frac{(-2\sin \left(A\right)\sin \left(x\right)+\cos \left(y\left(x\right)\right))\sin \left(y\left(x\right)\right)}{2}+\frac{\cos \left(x\right)\sin \left(x\right)}{2}+\frac{x}{2}+\mathit{\_}\mathit{C}\mathit{1}+\frac{y\left(x\right)}{2}=0\}$$ 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