85.33.68 problem 69

Internal problem ID [22691]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 69
Date solved : Thursday, October 02, 2025 at 09:06:44 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} i^{\prime }&=\frac {i t^{2}}{t^{3}-i^{3}} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 19

ode:=diff(i(t),t) = i(t)*t^2/(t^3-i(t)^3); 
dsolve(ode,i(t), singsol=all);
 

\[ i = {\left (-\frac {1}{\operatorname {LambertW}\left (-c_1 \,t^{3}\right )}\right )}^{{1}/{3}} t \]

✓ Mathematica. Time used: 4.238 (sec). Leaf size: 84

ode=D[i[t],{t,1}]== (i[t]*t^2)/(t^3-i[t]^3); 
ic={}; 
DSolve[{ode,ic},i[t],t,IncludeSingularSolutions->True]
 

\begin{align*} i(t)&\to -\frac {t}{\sqrt [3]{W\left (-e^{-3 c_1} t^3\right )}}\\ i(t)&\to \frac {\sqrt [3]{-1} t}{\sqrt [3]{W\left (-e^{-3 c_1} t^3\right )}}\\ i(t)&\to -\frac {(-1)^{2/3} t}{\sqrt [3]{W\left (-e^{-3 c_1} t^3\right )}}\\ i(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.493 (sec). Leaf size: 19

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

\[ i{\left (t \right )} = e^{C_{1} + \frac {W\left (- t^{3} e^{- 3 C_{1}}\right )}{3}} \]