74.7.21 problem 21

Internal problem ID [16019]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 21
Date solved : Thursday, March 13, 2025 at 07:16:29 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} t^{3}+y^{3}-t y^{2} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 56

ode:=t^3+y(t)^3-t*y(t)^2*diff(y(t),t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \left (3 \ln \left (t \right )+c_{1} \right )^{{1}/{3}} t \\ y &= -\frac {\left (3 \ln \left (t \right )+c_{1} \right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) t}{2} \\ y &= \frac {\left (3 \ln \left (t \right )+c_{1} \right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) t}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.18 (sec). Leaf size: 63

ode=( t^3+y[t]^3 )-( t*y[t]^2 )*D[y[t],t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to t \sqrt [3]{3 \log (t)+c_1} \\ y(t)\to -\sqrt [3]{-1} t \sqrt [3]{3 \log (t)+c_1} \\ y(t)\to (-1)^{2/3} t \sqrt [3]{3 \log (t)+c_1} \\ \end{align*}

✓ Sympy. Time used: 1.345 (sec). Leaf size: 68

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

\[ \left [ y{\left (t \right )} = \sqrt [3]{t^{3} \left (C_{1} + 3 \log {\left (t \right )}\right )}, \ y{\left (t \right )} = \frac {\sqrt [3]{t^{3} \left (C_{1} + 3 \log {\left (t \right )}\right )} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (t \right )} = \frac {\sqrt [3]{t^{3} \left (C_{1} + 3 \log {\left (t \right )}\right )} \left (-1 + \sqrt {3} i\right )}{2}\right ] \]