50.1.23 problem 2(i)

Internal problem ID [7795]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.2 THE NATURE OF SOLUTIONS. Page 9
Problem number : 2(i)
Date solved : Wednesday, March 05, 2025 at 05:06:11 AM
CAS classification : [_quadrature]

\begin{align*} \left (x^{3}+1\right ) y^{\prime }&=x \end{align*}

✓ Maple. Time used: 0.000 (sec). Leaf size: 39

ode:=(x^3+1)*diff(y(x),x) = x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\ln \left (x^{2}-x +1\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-1+2 x \right ) \sqrt {3}}{3}\right )}{3}-\frac {\ln \left (x +1\right )}{3}+c_{1} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 48

ode=(1+x^3)*D[y[x],x]==x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{6} \left (2 \sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )+\log \left (x^2-x+1\right )-2 \log (x+1)+6 c_1\right ) \]

✓ Sympy. Time used: 0.241 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + (x**3 + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} - \frac {\log {\left (x + 1 \right )}}{3} + \frac {\log {\left (x^{2} - x + 1 \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \left (2 x - 1\right )}{3} \right )}}{3} \]