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 : Saturday, March 29, 2025 at 04:34:57 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 71

ode:=3*y(x)^2*diff(y(x),x)+y(x)^3 = exp(-x); 
dsolve(ode,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*}

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

ode=3*y[x]^2*D[y[x],x]+y[x]^3==Exp[-x]; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.398 (sec). Leaf size: 58

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

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