Internal problem ID [8150]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 570.
ODE order: 1.
ODE degree: -1.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (\left (y^{\prime }\right )^{2}+1\right ) \left (\arctan \left (y^{\prime }\right )+a x \right )+y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 30
dsolve((diff(y(x),x)^2+1)*(arctan(diff(y(x),x))+a*x)+diff(y(x),x)=0,y(x), singsol=all)
\[ y \relax (x ) = \int \tan \left (\RootOf \left (a x \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z} +x a +\tan \left (\textit {\_Z} \right )+\textit {\_Z} \right )\right )d x +c_{1} \]
✓ Solution by Mathematica
Time used: 1.162 (sec). Leaf size: 58
DSolve[y'[x] + (a*x + ArcTan[y'[x]])*(1 + y'[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{y(x)=\frac {1}{a \left (K[1]^2+1\right )}+c_1,x=\frac {K[1]^2 (-\text {ArcTan}(K[1]))-\text {ArcTan}(K[1])-K[1]}{a \left (K[1]^2+1\right )}\right \},\{y(x),K[1]\}\right ] \]