73.1.8 problem 2 (ii)

Internal problem ID [19781]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 3. Solutions of first-order equations. Exercises at page 47
Problem number : 2 (ii)
Date solved : Thursday, October 02, 2025 at 04:43:20 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=b \,{\mathrm e}^{x} \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 14

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

\[ x = -\ln \left (-b t +{\mathrm e}^{-1}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 17

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

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

✓ Sympy. Time used: 0.101 (sec). Leaf size: 15

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

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