72.4.4 problem 8

Internal problem ID [14681]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.5 page 71
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 07:05:02 AM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} y \left (0\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 4.127 (sec). Leaf size: 131

dsolve([diff(y(t),t)=y(t)*(y(t)-1)*(y(t)-3),y(0) = -1],y(t), singsol=all)
 

\[ y = \frac {\left (2 \,{\mathrm e}^{6 t}-4\right ) \left (1-{\mathrm e}^{6 t}+\sqrt {{\mathrm e}^{6 t} \left ({\mathrm e}^{6 t}-2\right )}\right )^{{2}/{3}}+\left (\left (i \sqrt {3}-1\right ) \left (1-{\mathrm e}^{6 t}+\sqrt {{\mathrm e}^{6 t} \left ({\mathrm e}^{6 t}-2\right )}\right )^{{1}/{3}}-i \sqrt {3}-1\right ) \left ({\mathrm e}^{6 t}-\sqrt {{\mathrm e}^{6 t} \left ({\mathrm e}^{6 t}-2\right )}-2\right )}{\left (1-{\mathrm e}^{6 t}+\sqrt {{\mathrm e}^{6 t} \left ({\mathrm e}^{6 t}-2\right )}\right )^{{2}/{3}} \left (2 \,{\mathrm e}^{6 t}-4\right )} \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 104

DSolve[{D[y[t],t]==y[t]*(y[t]-1)*(y[t]-3),{y[0]==-1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {\sqrt [3]{2 \sqrt {e^{6 t} \left (e^{6 t}-2\right )^3}+8 e^{6 t}-2 e^{12 t}-8}}{e^{6 t}-2}-\frac {2^{2/3}}{\sqrt [3]{\sqrt {e^{6 t} \left (e^{6 t}-2\right )^3}+4 e^{6 t}-e^{12 t}-4}}+1 \]