4.15.41 $y^{\prime }(x)(a\cos (ay(x)+bx)-b\sin (ax+by(x)))-a\sin (ax+by(x))+b\cos (ay(x)+bx)=0$

ODE
$$y^{\prime }(x)(a\cos (ay(x)+bx)-b\sin (ax+by(x)))-a\sin (ax+by(x))+b\cos (ay(x)+bx)=0$$ ODE Classification

[_exact]

Book solution method
Exact equation

Mathematica ✓
cpu = 0.419222 (sec), leaf count = 26

$$\text{Solve}[\sin (ay(x)+bx)+\cos (ax+by(x))+c_1=0,y(x)]$$

Maple ✓
cpu = 0.17 (sec), leaf count = 23

$$\{\sin (bx+ay\left(x\right))+\cos (ax+by\left(x\right))+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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),'implicit')

Maple raw output

sin(b*x+a*y(x))+cos(a*x+b*y(x))+_C1 = 0