80.3.50 problem 53

Internal problem ID [21214]
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 3. First order nonlinear differential equations. Excercise 3.7 at page 67
Problem number : 53
Date solved : Thursday, October 02, 2025 at 07:26:53 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 21

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

\begin{align*} x &= t \left (\ln \left (t \right )-1\right ) \\ x &= t c_1 -{\mathrm e}^{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 26

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

\begin{align*} x(t)&\to c_1 t-e^{c_1}\\ x(t)&\to t (\log (t)-1) \end{align*}

✓ Sympy. Time used: 15.526 (sec). Leaf size: 19

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

\[ C_{1} - W\left (- \frac {e^{\frac {x{\left (t \right )}}{t}}}{t}\right ) + \frac {x{\left (t \right )}}{t} = 0 \]