76.5.17 problem 17

Internal problem ID [17437]
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 : Tuesday, January 28, 2025 at 10:35:44 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 79

dsolve(5*(1+t^2)*diff(y(t),t)=4*t*y(t)*(y(t)^3-1),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*}

✓ Solution by Mathematica

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

DSolve[5*(1+t^2)*D[y[t],t]==4*t*y[t]*(y[t]^3-1),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*}