83.5.13 problem 13

Internal problem ID [19029]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (D) at page 16
Problem number : 13
Date solved : Thursday, March 13, 2025 at 01:24:35 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 18

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

\[ y \left (x \right ) = \tan \left (\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-x -1}\right )+x +1\right ) \]

✓ Mathematica. Time used: 60.117 (sec). Leaf size: 21

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

\[ y(x)\to \tan \left (W\left (c_1 \left (-e^{-x-1}\right )\right )+x+1\right ) \]

✓ Sympy. Time used: 3.041 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = \tan {\left (x + W\left (C_{1} e^{- x - 1}\right ) + 1 \right )} \]