63.1.12 problem 12

Internal problem ID [15452]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 12
Date solved : Thursday, October 02, 2025 at 10:14:58 AM
CAS classification : [_separable]

\begin{align*} \left (t^{2}+x t^{2}\right ) x^{\prime }+x^{2}+t x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 16

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

\[ x = \frac {1}{\operatorname {LambertW}\left (c_1 t \,{\mathrm e}^{-\frac {1}{t}}\right )} \]

✓ Mathematica. Time used: 2.053 (sec). Leaf size: 28

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

\begin{align*} x(t)&\to \frac {1}{W\left (t e^{-\frac {1}{t}-2-c_1}\right )}\\ x(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.572 (sec). Leaf size: 22

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

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