80.3.37 problem 40

Internal problem ID [21201]
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 : 40
Date solved : Thursday, October 02, 2025 at 07:26:20 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 16

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

\[ x = \sqrt {\frac {1}{\operatorname {LambertW}\left (c_1 \,t^{2}\right )}}\, t \]

✓ Mathematica. Time used: 5.483 (sec). Leaf size: 49

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

\begin{align*} x(t)&\to -\frac {t}{\sqrt {W\left (e^{-2 c_1} t^2\right )}}\\ x(t)&\to \frac {t}{\sqrt {W\left (e^{-2 c_1} t^2\right )}}\\ x(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.678 (sec). Leaf size: 17

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

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