44.1.30 problem 3(f)

Internal problem ID [9088]
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 : 3(f)
Date solved : Tuesday, September 30, 2025 at 06:03:53 PM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 24

ode:=(1+x)*(x^2+1)*diff(y(x),x) = 2*x^2+x; 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {3 \ln \left (x^{2}+1\right )}{4}-\frac {\arctan \left (x \right )}{2}+\frac {\ln \left (x +1\right )}{2}+1 \]

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

ode=(x+1)*(x^2+1)*D[y[x],x]==2*x^2+x; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _0^x\frac {K[1] (2 K[1]+1)}{K[1]^3+K[1]^2+K[1]+1}dK[1]+1 \end{align*}

✓ Sympy. Time used: 0.141 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = \frac {\log {\left (x + 1 \right )}}{2} + \frac {3 \log {\left (x^{2} + 1 \right )}}{4} - \frac {\operatorname {atan}{\left (x \right )}}{2} + 1 \]