80.3.48 problem 51

Internal problem ID [21212]
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 : 51
Date solved : Thursday, October 02, 2025 at 07:26:51 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {x^{\prime }}^{2}-x t +x&=0 \end{align*}

✓ Maple. Time used: 0.093 (sec). Leaf size: 71

ode:=diff(x(t),t)^2-t*x(t)+x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x &= 0 \\ x &= \frac {t^{3}}{9}+\frac {c_1^{2}}{4}+\frac {c_1 \sqrt {\left (t -1\right )^{3}}}{3}-\frac {t^{2}}{3}+\frac {t}{3}-\frac {1}{9} \\ x &= \frac {t^{3}}{9}+\frac {c_1^{2}}{4}-\frac {c_1 \sqrt {\left (t -1\right )^{3}}}{3}-\frac {t^{2}}{3}+\frac {t}{3}-\frac {1}{9} \\ \end{align*}

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 110

ode=D[x[t],t]^2-t*x[t]+x[t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{36} \left (4 t^3-12 t^2-12 t \left (-1+c_1 \sqrt {t-1}\right )+12 c_1 \sqrt {t-1}-4+9 c_1{}^2\right )\\ x(t)&\to \frac {1}{36} \left (4 t^3-12 t^2+12 t \left (1+c_1 \sqrt {t-1}\right )-12 c_1 \sqrt {t-1}-4+9 c_1{}^2\right )\\ x(t)&\to 0 \end{align*}

✗ Sympy

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

TypeError : cannot determine truth value of Relational: _X0*(-288*t**3 + 864*t**2 - 864*t + 288) < -72*C1**2*t**3 + 216*C1**2*t**2 - 216*C1**2*t + 72*C1**2 + 32*t**6 - 192*t**5 + 480*t**4 - 640*t**3 + 480*t**2 - 192*t + 32