23.2.348 problem 397

Internal problem ID [5703]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 397
Date solved : Tuesday, September 30, 2025 at 02:00:46 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*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 20

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

\[ y = \int \tan \left (\operatorname {RootOf}\left (a x +\sin \left (\textit {\_Z} \right ) \cos \left (\textit {\_Z} \right )+\textit {\_Z} \right )\right )d x +c_1 \]

✓ Mathematica. Time used: 0.738 (sec). Leaf size: 58

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq((a*x + atan(Derivative(y(x), x)))*(Derivative(y(x), x)**2 + 1) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : multiple generators [_X0, atan(_X0)] 
No algorithms are implemented to solve equation _X0**2*a*x + _X0**2*atan(_X0) + _X0 + a*x + atan(_X0)