62.12.15 problem Ex 16

Internal problem ID [12790]
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 : Monday, March 31, 2025 at 07:07:44 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: 0.193 (sec). Leaf size: 69

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

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

✓ Sympy. Time used: 1.258 (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)
 

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