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 = 0.419222 (sec), leaf count = 26

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

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

\[ \left \{ \sin \left (bx+ay \relax (x ) \right ) +\cos \left (ax+by \relax (x ) \right ) +{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

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