Internal problem ID [1049]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {\left (y^{3}-1\right ) {\mathrm e}^{x}+3 y^{2} \left (1+{\mathrm e}^{x}\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 91
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 \left (x \right ) = \frac {{\left (\left ({\mathrm e}^{x}-1\right ) \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{\frac {1}{3}}}{1+{\mathrm e}^{x}} y \left (x \right ) = \frac {\left (i \sqrt {3}-1\right ) {\left (\left ({\mathrm e}^{x}-1\right ) \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{\frac {1}{3}}}{2+2 \,{\mathrm e}^{x}} y \left (x \right ) = -\frac {\left (1+i \sqrt {3}\right ) {\left (\left ({\mathrm e}^{x}-1\right ) \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{\frac {1}{3}}}{2+2 \,{\mathrm e}^{x}} \end{align*}
✓ Solution by Mathematica
Time used: 0.984 (sec). Leaf size: 77
DSolve[{((y[x]^3-1)*Exp[x])+(3*y[x]^2*(Exp[x]+1))*y'[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*}