4.9.43 \((y(x)+x) y'(x)+\tan (y(x))=0\)

ODE
\[ (y(x)+x) y'(x)+\tan (y(x))=0 \] ODE Classification

[[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0423392 (sec), leaf count = 17

\[\text {Solve}\left [y(x)+\cot (y(x))+x=c_1 \csc (y(x)),y(x)\right ]\]

Maple
cpu = 0.011 (sec), leaf count = 24

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

DSolve[Tan[y[x]] + (x + y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[x + Cot[y[x]] + y[x] == C[1]*Csc[y[x]], y[x]]

Maple raw input

dsolve((x+y(x))*diff(y(x),x)+tan(y(x)) = 0, y(x),'implicit')

Maple raw output

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