76.5.14 problem 14

Internal problem ID [17434]
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 : 14
Date solved : Tuesday, January 28, 2025 at 10:35:38 AM
CAS classification : [_Bernoulli]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 127

dsolve(diff(y(t),t)=y(t)*(t*y(t)^3-1),y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 9.983 (sec). Leaf size: 124

DSolve[D[y[t],t]==y[t]*(t*y[t]^3-1),y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to \frac {1}{\sqrt [3]{e^{3 t} \left (-3 \int _1^te^{-3 K[1]} K[1]dK[1]+c_1\right )}} \\ y(t)\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{e^{3 t} \left (-3 \int _1^te^{-3 K[1]} K[1]dK[1]+c_1\right )}} \\ y(t)\to \frac {(-1)^{2/3}}{\sqrt [3]{e^{3 t} \left (-3 \int _1^te^{-3 K[1]} K[1]dK[1]+c_1\right )}} \\ y(t)\to 0 \\ \end{align*}