56.12.15 problem Ex 16

Internal problem ID [14142]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 19. Summary. Page 29
Problem number : Ex 16
Date solved : Thursday, October 02, 2025 at 09:17:01 AM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 15

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

\[ y = \arctan \left (x \right )-1+{\mathrm e}^{-\arctan \left (x \right )} c_1 \]

✓ Mathematica. Time used: 0.129 (sec). Leaf size: 18

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

\begin{align*} y(x)&\to \arctan (x)+c_1 e^{-\arctan (x)}-1 \end{align*}

✓ Sympy. Time used: 0.762 (sec). Leaf size: 14

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

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