12.6.20 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 : Monday, January 27, 2025 at 05:30:48 AM
CAS classification : [_separable]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.118 (sec). Leaf size: 89

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

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

✓ Solution by Mathematica

Time used: 0.934 (sec). Leaf size: 77

DSolve[{((y[x]^3-1)*Exp[x])+(3*y[x]^2*(Exp[x]+1))*D[y[x],x]==0,y[0]==0},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*}