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

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

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.0768081 (sec), leaf count = 16

\[\left \{\left \{y(x)\to \sin ^{-1}\left (\frac {c_1}{x}\right )-x\right \}\right \}\]

Maple
cpu = 0.081 (sec), leaf count = 59

\[ \left \{ {\frac { \left (-{\it \_C1}\,{x}^{2}+ \left (\tan \relax (x ) \right ) ^{2}+1 \right ) \left (\tan \left (y \relax (x ) \right ) \right ) ^{2}-2\,\tan \relax (x ) \tan \left (y \relax (x ) \right ) {\it \_C1}\,{x}^{2}+1+ \left (-{\it \_C1}\,{x}^{2}+1 \right ) \left (\tan \relax (x ) \right ) ^{2}}{ \left (\tan \relax (x ) +\tan \left (y \relax (x ) \right ) \right ) ^{2}{x}^{2}}}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> -x + ArcSin[C[1]/x]}}

Maple raw input

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

Maple raw output

((-_C1*x^2+tan(x)^2+1)*tan(y(x))^2-2*tan(x)*tan(y(x))*_C1*x^2+1+(-_C1*x^2+1)*tan
(x)^2)/(tan(x)+tan(y(x)))^2/x^2 = 0