76.5.17 problem 17

Internal problem ID [17366]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 17
Date solved : Monday, March 31, 2025 at 04:07:55 PM
CAS classification : [_separable]

\begin{align*} 5 \left (t^{2}+1\right ) y^{\prime }&=4 t y \left (y^{3}-1\right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 79

ode:=5*(t^2+1)*diff(y(t),t) = 4*t*y(t)*(y(t)^3-1); 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \frac {1}{\left (c_1 \,t^{2} \left (t^{2}+1\right )^{{1}/{5}}+\left (t^{2}+1\right )^{{1}/{5}} c_1 +1\right )^{{1}/{3}}} \\ y &= -\frac {1+i \sqrt {3}}{2 {\left (\left (t^{2}+1\right )^{{6}/{5}} c_1 +1\right )}^{{1}/{3}}} \\ y &= \frac {i \sqrt {3}-1}{2 {\left (\left (t^{2}+1\right )^{{6}/{5}} c_1 +1\right )}^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 0.362 (sec). Leaf size: 79

ode=5*(1+t^2)*D[y[t],t]==4*t*y[t]*(y[t]^3-1); 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1] \left (K[1]^2+K[1]+1\right )}dK[1]\&\right ]\left [\frac {2}{5} \log \left (t^2+1\right )+c_1\right ] \\ y(t)\to 0 \\ y(t)\to 1 \\ y(t)\to -\sqrt [3]{-1} \\ y(t)\to (-1)^{2/3} \\ \end{align*}

✓ Sympy. Time used: 0.308 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*t*(y(t)**3 - 1)*y(t) + (5*t**2 + 5)*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ - \frac {2 \log {\left (t^{2} + 1 \right )}}{5} + \frac {\log {\left (y^{3}{\left (t \right )} - 1 \right )}}{3} - \log {\left (y{\left (t \right )} \right )} = C_{1} \]