80.2.5 problem 6

Internal problem ID [21148]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 2. Theory of first order differential equations. Excercise 2.6 at page 37
Problem number : 6
Date solved : Thursday, October 02, 2025 at 07:10:19 PM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.067 (sec). Leaf size: 24

ode:=exp(diff(x(t),t)) = x(t); 
ic:=[x(t__0) = a]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = {\mathrm e}^{\operatorname {RootOf}\left (t +\operatorname {Ei}_{1}\left (-\textit {\_Z} \right )-t_{0} -\operatorname {Ei}_{1}\left (-\ln \left (a \right )\right )\right )} \]

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 14

ode=Exp[D[x[t],t]]==x[t]; 
ic={x[t0]==a}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \text {LogIntegral}^{(-1)}(\operatorname {LogIntegral}(a)+t-\text {t0}) \end{align*}

✓ Sympy. Time used: 0.182 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
a = symbols("a") 
t0 = symbols("t0") 
x = Function("x") 
ode = Eq(-x(t) + exp(Derivative(x(t), t)),0) 
ics = {x(t0): a} 
dsolve(ode,func=x(t),ics=ics)
 

\[ \operatorname {li}{\left (x{\left (t \right )} \right )} = t - t_{0} + \operatorname {li}{\left (a \right )} \]