78.1.16 problem 3

Internal problem ID [20942]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 1, First order ODEs. Problems section 1.5
Problem number : 3
Date solved : Thursday, October 02, 2025 at 06:49:50 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {x}{y}-\frac {x}{1+y} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.193 (sec). Leaf size: 60

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

\[ y = \frac {\left (9+6 x^{2}+2 \sqrt {9 x^{4}+27 x^{2}+20}\right )^{{1}/{3}}}{2}+\frac {1}{2 \left (9+6 x^{2}+2 \sqrt {9 x^{4}+27 x^{2}+20}\right )^{{1}/{3}}}-\frac {1}{2} \]

✓ Mathematica. Time used: 1.992 (sec). Leaf size: 69

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

\begin{align*} y(x)&\to \frac {1}{2} \left (\sqrt [3]{6 x^2+2 \sqrt {9 x^4+27 x^2+20}+9}+\frac {1}{\sqrt [3]{6 x^2+2 \sqrt {9 x^4+27 x^2+20}+9}}-1\right ) \end{align*}

✓ Sympy. Time used: 162.339 (sec). Leaf size: 117

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

\[ y{\left (x \right )} = \frac {\sqrt [3]{- 162 x^{2} + 4 \sqrt {\frac {6561 x^{4}}{4} + \frac {19683 x^{2}}{4} + 3645} - 243}}{12} - \frac {\sqrt {3} i \sqrt [3]{- 162 x^{2} + 4 \sqrt {\frac {6561 x^{4}}{4} + \frac {19683 x^{2}}{4} + 3645} - 243}}{12} - \frac {1}{2} - \frac {3}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{- 162 x^{2} + 4 \sqrt {\frac {6561 x^{4}}{4} + \frac {19683 x^{2}}{4} + 3645} - 243}} \]