ODE
\[ \left (y'(x)^2+1\right ) \left (a x+\tan ^{-1}\left (y'(x)\right )\right )+y'(x)=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(x\)
Mathematica ✗
cpu = 0.105349 (sec), leaf count = 0 , could not solve
DSolve[Derivative[1][y][x] + (a*x + ArcTan[Derivative[1][y][x]])*(1 + Derivative[1][y][x]^2) == 0, y[x], x]
Maple ✓
cpu = 0.074 (sec), leaf count = 30
\[ \left \{ y \left ( x \right ) =\int \!\tan \left ( {\it RootOf} \left ( ax \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}+ \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}{\it \_Z}+ax+\tan \left ( {\it \_Z} \right ) +{\it \_Z} \right ) \right ) \,{\rm d}x+{\it \_C1} \right \} \] Mathematica raw input
DSolve[y'[x] + (a*x + ArcTan[y'[x]])*(1 + y'[x]^2) == 0,y[x],x]
Mathematica raw output
DSolve[Derivative[1][y][x] + (a*x + ArcTan[Derivative[1][y][x]])*(1 + Derivative[1][y][x]^2) == 0, y[x], x]
Maple raw input
dsolve((1+diff(y(x),x)^2)*(arctan(diff(y(x),x))+a*x)+diff(y(x),x) = 0, y(x),'implicit')
Maple raw output
y(x) = Int(tan(RootOf(a*x*tan(_Z)^2+tan(_Z)^2*_Z+a*x+tan(_Z)+_Z)),x)+_C1