##### 4.15.41 $$y'(x) (a \cos (a y(x)+b x)-b \sin (a x+b y(x)))-a \sin (a x+b y(x))+b \cos (a y(x)+b x)=0$$

ODE
$y'(x) (a \cos (a y(x)+b x)-b \sin (a x+b y(x)))-a \sin (a x+b y(x))+b \cos (a y(x)+b x)=0$ ODE Classiﬁcation

[_exact]

Book solution method
Exact equation

Mathematica
cpu = 1.32205 (sec), leaf count = 26

$\text {Solve}[\sin (a y(x)+b x)+\cos (a x+b y(x))+c_1=0,y(x)]$

Maple
cpu = 0.19 (sec), leaf count = 40

$\left [y \left (x \right ) = \frac {-b x +\RootOf \left (a^{2} x -b^{2} x -a \pi +a \arccos \left (\sin \left (\textit {\_Z} \right )+\textit {\_C1} \right )+\textit {\_Z} b \right )}{a}\right ]$ Mathematica raw input

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

Mathematica raw output

Solve[C[1] + Cos[a*x + b*y[x]] + Sin[b*x + a*y[x]] == 0, y[x]]

Maple raw input

dsolve((a*cos(b*x+a*y(x))-b*sin(a*x+b*y(x)))*diff(y(x),x)+b*cos(b*x+a*y(x))-a*sin(a*x+b*y(x)) = 0, y(x))

Maple raw output

[y(x) = (-b*x+RootOf(a^2*x-b^2*x-a*Pi+a*arccos(sin(_Z)+_C1)+_Z*b))/a]