34.4.16 problem 19 (r)

Internal problem ID [7943]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 6. Equations of first order and first degree (Linear equations). Supplemetary problems. Page 39
Problem number : 19 (r)
Date solved : Tuesday, September 30, 2025 at 05:11:01 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: 0.006 (sec). Leaf size: 18

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

\[ y = \tan \left (\operatorname {LambertW}\left (-c_1 \,{\mathrm e}^{-x -1}\right )+x +1\right ) \]

✓ Mathematica. Time used: 0.151 (sec). Leaf size: 95

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

\[ \text {Solve}\left [x=\exp \left (\int _1^{y(x)}\frac {1}{-K[1]^2-1}dK[1]\right ) \int _1^{y(x)}-\frac {\exp \left (-\int _1^{K[2]}\frac {1}{-K[1]^2-1}dK[1]\right ) \arctan (K[2])}{-K[2]^2-1}dK[2]+c_1 \exp \left (\int _1^{y(x)}\frac {1}{-K[1]^2-1}dK[1]\right ),y(x)\right ] \]

✓ Sympy. Time used: 1.925 (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)
 

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