29.37.19 problem 1140

Internal problem ID [5677]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1140
Date solved : Monday, January 27, 2025 at 01:06:34 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: 0.010 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

Time used: 1.207 (sec). Leaf size: 58

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

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