7.5.26 problem 26

Internal problem ID [130]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.6 (substitution and exact equations). Problems at page 72
Problem number : 26
Date solved : Friday, February 07, 2025 at 07:55:56 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

\begin{align*} 3 y^{\prime } y^{2}+y^{3}&={\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 71

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

\begin{align*} y &= {\mathrm e}^{-x} \left ({\mathrm e}^{2 x} \left (x +c_1 \right )\right )^{{1}/{3}} \\ y &= -\frac {\left ({\mathrm e}^{2 x} \left (x +c_1 \right )\right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) {\mathrm e}^{-x}}{2} \\ y &= \frac {\left ({\mathrm e}^{2 x} \left (x +c_1 \right )\right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) {\mathrm e}^{-x}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.392 (sec). Leaf size: 72

DSolve[3*y[x]^2*D[y[x],x]+y[x]^3==Exp[-x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to e^{-x/3} \sqrt [3]{x+c_1} \\ y(x)\to -\sqrt [3]{-1} e^{-x/3} \sqrt [3]{x+c_1} \\ y(x)\to (-1)^{2/3} e^{-x/3} \sqrt [3]{x+c_1} \\ \end{align*}