72.4.1 problem 5

Internal problem ID [14678]
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 : 5
Date solved : Tuesday, January 28, 2025 at 06:47:29 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 )&=4 \end{align*}

✓ Solution by Maple

Time used: 4.700 (sec). Leaf size: 133

dsolve([diff(y(t),t)=y(t)*(y(t)-1)*(y(t)-3),y(0) = 4],y(t), singsol=all)
 

\[ y = \frac {48 \left (\frac {{\mathrm e}^{6 t}}{3}-\frac {9}{16}\right ) \left (27-32 \,{\mathrm e}^{6 t}+8 \sqrt {16 \,{\mathrm e}^{12 t}-27 \,{\mathrm e}^{6 t}}\right )^{{2}/{3}}+48 \left ({\mathrm e}^{6 t}-\frac {\sqrt {16 \,{\mathrm e}^{12 t}-27 \,{\mathrm e}^{6 t}}}{4}-\frac {27}{16}\right ) \left (\left (27-32 \,{\mathrm e}^{6 t}+8 \sqrt {16 \,{\mathrm e}^{12 t}-27 \,{\mathrm e}^{6 t}}\right )^{{1}/{3}}+3\right )}{\left (27-32 \,{\mathrm e}^{6 t}+8 \sqrt {16 \,{\mathrm e}^{12 t}-27 \,{\mathrm e}^{6 t}}\right )^{{2}/{3}} \left (16 \,{\mathrm e}^{6 t}-27\right )} \]

✓ Solution by Mathematica

Time used: 0.095 (sec). Leaf size: 132

DSolve[{D[y[t],t]==y[t]*(y[t]-1)*(y[t]-3),{y[0]==4}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {3 i \left (\sqrt {3}+i\right ) \sqrt [3]{4 \sqrt {e^{6 t} \left (16 e^{6 t}-27\right )^3}+864 e^{6 t}-256 e^{12 t}-729}}{32 e^{6 t}-54}+\frac {9 \left (1+i \sqrt {3}\right )}{2 \sqrt [3]{4 \sqrt {e^{6 t} \left (16 e^{6 t}-27\right )^3}+864 e^{6 t}-256 e^{12 t}-729}}+1 \]