30.5.6 problem 6

Internal problem ID [7505]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.6, Substitutions and Transformations. Exercises. page 76
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 04:40:36 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Bernoulli]

\begin{align*} y \,{\mathrm e}^{-2 x}+y^{3}-{\mathrm e}^{-2 x} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 69

ode:=y(x)*exp(-2*x)+y(x)^3-exp(-2*x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {2}\, \sqrt {2 \,{\mathrm e}^{-6 x} c_1 -{\mathrm e}^{-2 x}}}{2 c_1 \,{\mathrm e}^{-4 x}-1} \\ y &= \frac {\sqrt {2}\, \sqrt {2 \,{\mathrm e}^{-6 x} c_1 -{\mathrm e}^{-2 x}}}{-2 c_1 \,{\mathrm e}^{-4 x}+1} \\ \end{align*}

✓ Mathematica. Time used: 0.525 (sec). Leaf size: 66

ode=( y[x]*Exp[-2*x]+y[x]^3)-(Exp[-2*x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {i \sqrt {2} e^x}{\sqrt {e^{4 x}-2 c_1}}\\ y(x)&\to \frac {i \sqrt {2} e^x}{\sqrt {e^{4 x}-2 c_1}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.602 (sec). Leaf size: 46

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

\[ \left [ y{\left (x \right )} = - \sqrt {2} \sqrt {\frac {e^{2 x}}{C_{1} - e^{4 x}}}, \ y{\left (x \right )} = \sqrt {2} \sqrt {\frac {e^{2 x}}{C_{1} - e^{4 x}}}\right ] \]