80.3.49 problem 52

Internal problem ID [21213]
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 : 52
Date solved : Thursday, October 02, 2025 at 07:26:52 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} x&=x^{\prime } t -{x^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 19

ode:=x(t) = t*diff(x(t),t)-diff(x(t),t)^2; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x &= \frac {t^{2}}{4} \\ x &= c_1 \left (-c_1 +t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 25

ode=x[t]==t*D[x[t],t]-D[x[t],t]^2; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to c_1 (t-c_1)\\ x(t)&\to \frac {t^2}{4} \end{align*}

✓ Sympy. Time used: 1.173 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t*Derivative(x(t), t) + x(t) + Derivative(x(t), t)**2,0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {t^{2}}{4} - \frac {\left (C_{1} + t\right )^{2}}{4} \]