6.6.20 problem 20

Internal problem ID [1699]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Exact equations. Section 2.5 Page 79
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 05:15:10 AM
CAS classification : [_separable]

\begin{align*} \left (y^{3}-1\right ) {\mathrm e}^{x}+3 y^{2} \left ({\mathrm e}^{x}+1\right ) y^{\prime }&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.160 (sec). Leaf size: 89

ode:=(y(x)^3-1)*exp(x)+3*y(x)^2*(exp(x)+1)*diff(y(x),x) = 0; 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= \frac {{\left (\left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )^{2}\right )}^{{1}/{3}}}{{\mathrm e}^{x}+1} \\ y &= \frac {{\left (\left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 \,{\mathrm e}^{x}+2} \\ y &= -\frac {{\left (\left ({\mathrm e}^{x}-1\right ) \left ({\mathrm e}^{x}+1\right )^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 \,{\mathrm e}^{x}+2} \\ \end{align*}

✓ Mathematica. Time used: 0.713 (sec). Leaf size: 77

ode=((y[x]^3-1)*Exp[x])+(3*y[x]^2*(Exp[x]+1))*D[y[x],x]==0; 
ic=y[0]==0; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt [3]{\frac {e^x-1}{e^x+1}}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{\frac {e^x-1}{e^x+1}}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{\frac {e^x-1}{e^x+1}} \end{align*}

✓ Sympy. Time used: 1.503 (sec). Leaf size: 63

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((y(x)**3 - 1)*exp(x) + (3*exp(x) + 3)*y(x)**2*Derivative(y(x), x),0) 
ics = {y(0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{1 - \frac {2}{e^{x} + 1}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{1 - \frac {2}{e^{x} + 1}}}{2}, \ y{\left (x \right )} = \sqrt [3]{1 - \frac {2}{e^{x} + 1}}\right ] \]