68.1.34 problem 41

Internal problem ID [17101]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 41
Date solved : Thursday, October 02, 2025 at 01:43:13 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 14

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

\[ y = \arctan \left (x \right )-\ln \left (x +1\right )+c_1 \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to \int _1^x-\frac {(K[1]-1) K[1]}{K[1]^3+K[1]^2+K[1]+1}dK[1]+c_1 \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 12

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

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