30.3.23 problem 24

Internal problem ID [7479]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.4, Exact equations. Exercises. page 64
Problem number : 24
Date solved : Tuesday, September 30, 2025 at 04:38:11 PM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} x \left (1\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 17

ode:=exp(t)*x(t)+1+(exp(t)-1)*diff(x(t),t) = 0; 
ic:=[x(1) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {-t +{\mathrm e}}{{\mathrm e}^{t}-1} \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 18

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

\begin{align*} x(t)&\to \frac {e-t}{e^t-1} \end{align*}

✓ Sympy. Time used: 0.179 (sec). Leaf size: 12

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

\[ x{\left (t \right )} = \frac {e - t}{e^{t} - 1} \]