8.5.26 problem 26

Internal problem ID [754]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 26
Date solved : Wednesday, February 05, 2025 at 03:58:13 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 71

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

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

✓ Solution by Mathematica

Time used: 0.356 (sec). Leaf size: 72

DSolve[y[x]^3+3*y[x]^2*D[y[x],x] == 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*}