60.1.567 problem 570

Internal problem ID [10581]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 570
Date solved : Monday, January 27, 2025 at 09:16:39 PM
CAS classification : [_quadrature]

\begin{align*} \left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+a x \right )+y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 20

dsolve((diff(y(x),x)^2+1)*(arctan(diff(y(x),x))+a*x)+diff(y(x),x)=0,y(x), singsol=all)
 

\[ y = \int \tan \left (\operatorname {RootOf}\left (a x +\cos \left (\textit {\_Z} \right ) \sin \left (\textit {\_Z} \right )+\textit {\_Z} \right )\right )d x +c_{1} \]

✓ Solution by Mathematica

Time used: 1.193 (sec). Leaf size: 58

DSolve[D[y[x],x] + (a*x + ArcTan[D[y[x],x]])*(1 + D[y[x],x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\left \{y(x)=\frac {1}{a \left (K[1]^2+1\right )}+c_1,x=\frac {K[1]^2 (-\arctan (K[1]))-\arctan (K[1])-K[1]}{a \left (K[1]^2+1\right )}\right \},\{y(x),K[1]\}\right ] \]