80.3.51 problem 54

Internal problem ID [21215]
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 : 54
Date solved : Thursday, October 02, 2025 at 07:26:55 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} x&=x^{\prime } t -\ln \left (x^{\prime }\right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 23

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

\begin{align*} x &= -\ln \left (\frac {1}{t}\right )+1 \\ x &= t c_1 -\ln \left (c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 27

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

\begin{align*} x(t)&\to c_1 t-\log (c_1)\\ x(t)&\to 1-\log \left (\frac {1}{t}\right ) \end{align*}

✓ Sympy. Time used: 0.656 (sec). Leaf size: 15

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

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