72.4.3 problem 7

Internal problem ID [14680]
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 : 7
Date solved : Tuesday, January 28, 2025 at 06:58:49 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 )&=2 \end{align*}

✓ Solution by Maple

Time used: 32.958 (sec). Leaf size: 219

dsolve([diff(y(t),t)=y(t)*(y(t)-1)*(y(t)-3),y(0) = 2],y(t), singsol=all)
 

\[ y = \frac {\left (16 \,{\mathrm e}^{6 t}+9\right ) \left (1+8 \,{\mathrm e}^{6 t}+4 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}\right )^{{2}/{3}}+\left (24 \,{\mathrm e}^{6 t}+12 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}+9\right ) \left (1+8 \,{\mathrm e}^{6 t}+4 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}\right )^{{1}/{3}}+48 \,{\mathrm e}^{6 t}+24 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}+9}{\left (16 \,{\mathrm e}^{6 t}+3\right ) \left (1+8 \,{\mathrm e}^{6 t}+4 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}\right )^{{2}/{3}}+\left (8 \,{\mathrm e}^{6 t}+4 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}+3\right ) \left (1+8 \,{\mathrm e}^{6 t}+4 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}\right )^{{1}/{3}}+16 \,{\mathrm e}^{6 t}+8 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}+3} \]

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 105

DSolve[{D[y[t],t]==y[t]*(y[t]-1)*(y[t]-3),{y[0]==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {\sqrt [3]{2 \sqrt {e^{6 t} \left (4 e^{6 t}+1\right )^3}+8 e^{6 t}+16 e^{12 t}+1}}{4 e^{6 t}+1}+\frac {1}{\sqrt [3]{2 \sqrt {e^{6 t} \left (4 e^{6 t}+1\right )^3}+8 e^{6 t}+16 e^{12 t}+1}}+1 \]