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 = 1.97752 (sec), leaf count = 45
\[\text {Solve}\left [\left \{\frac {1}{a K[1]^2+a}+c_1=y(x),x=-\frac {\frac {K[1]}{K[1]^2+1}+\tan ^{-1}(K[1])}{a}\right \},\{y(x),K[1]\}\right ]\]
Maple ✓
cpu = 0.339 (sec), leaf count = 30
\[[y \left (x \right ) = \int \tan \left (\RootOf \left (a x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z} +a x +\tan \left (\textit {\_Z} \right )+\textit {\_Z} \right )\right )d x +\textit {\_C1}]\] Mathematica raw input
DSolve[y'[x] + (a*x + ArcTan[y'[x]])*(1 + y'[x]^2) == 0,y[x],x]
Mathematica raw output
Solve[{C[1] + (a + a*K[1]^2)^(-1) == y[x], x == -((ArcTan[K[1]] + K[1]/(1 + K[1]^2))/a)}, {y[x], K[1]}]
Maple raw input
dsolve((1+diff(y(x),x)^2)*(arctan(diff(y(x),x))+a*x)+diff(y(x),x) = 0, y(x))
Maple raw output
[y(x) = Int(tan(RootOf(a*x*tan(_Z)^2+tan(_Z)^2*_Z+a*x+tan(_Z)+_Z)),x)+_C1]