8.5.38 problem 38

Internal problem ID [766]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 38
Date solved : Saturday, March 29, 2025 at 10:21:12 PM
CAS classification : [_exact]

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 22

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

\[ y = \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} x +x^{2}-2 \ln \left (\cos \left (\textit {\_Z} \right )\right )+2 c_1 \right )\right ) \]

✓ Mathematica. Time used: 0.146 (sec). Leaf size: 30

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x*y(x)**2 - x - y(x)**2*atan(y(x)) - atan(y(x)))/(x + y(x)) cannot be solved by the factorable group method