89.1.23 problem 23

Internal problem ID [24258]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 21
Problem number : 23
Date solved : Thursday, October 02, 2025 at 10:02:12 PM
CAS classification : [_separable]

\begin{align*} y-\left ({\mathrm e}^{3 x}+1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 26

ode:=y(x)-(exp(3*x)+1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{2 x}-{\mathrm e}^{x}+1\right )^{{1}/{3}} \left ({\mathrm e}^{x}+1\right )^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 29

ode=y[x]-(Exp[3*x]+1)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^{-\frac {2}{3} \text {arctanh}\left (2 e^{3 x}+1\right )}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.182 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(exp(3*x) + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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