80.2.18 problem 25

Internal problem ID [21161]
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 : 25
Date solved : Thursday, October 02, 2025 at 07:15:03 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 34

ode:=diff(x(t),t) = exp(x(t))-t; 
dsolve(ode,x(t), singsol=all);
 

\[ x = -\frac {t^{2}}{2}+\ln \left (2\right )-\ln \left (-\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, t}{2}\right )-2 c_1 \right ) \]

✓ Mathematica. Time used: 0.413 (sec). Leaf size: 38

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

\begin{align*} x(t)&\to -\frac {t^2}{2}-\log \left (-\sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {t}{\sqrt {2}}\right )+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.803 (sec). Leaf size: 27

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

\[ x{\left (t \right )} = \log {\left (\frac {1}{\left (C_{1} - \int \frac {1}{\sqrt {e^{t^{2}}}}\, dt\right ) \sqrt {e^{t^{2}}}} \right )} \]