80.3.44 problem 47

Internal problem ID [21208]
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 : 47
Date solved : Thursday, October 02, 2025 at 07:26:43 PM
CAS classification : [_separable]

\begin{align*} x^{\prime }+2 x t&=-4 t x^{3} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

ode:=diff(x(t),t)+2*t*x(t) = -4*t*x(t)^3; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x &= \frac {1}{\sqrt {{\mathrm e}^{2 t^{2}} c_1 -2}} \\ x &= -\frac {1}{\sqrt {{\mathrm e}^{2 t^{2}} c_1 -2}} \\ \end{align*}

✓ Mathematica. Time used: 29.138 (sec). Leaf size: 91

ode=D[x[t],t]+2*t*x[t]==-4*t*x[t]^3; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.733 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*t*x(t)**3 + 2*t*x(t) + Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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