44.4.44 problem 26

Internal problem ID [7057]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.1 Solution curves without a solution. Exercises 2.1 at page 44
Problem number : 26
Date solved : Monday, January 27, 2025 at 02:42:47 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y \left (2-y\right ) \left (4-y\right ) \end{align*}

✓ Solution by Maple

Time used: 0.165 (sec). Leaf size: 73

dsolve(diff(y(x),x)=y(x)*(2-y(x))*(4-y(x)),y(x), singsol=all)
 

\begin{align*} y &= -\frac {2 \,{\mathrm e}^{8 x} c_{1}}{\sqrt {-{\mathrm e}^{8 x} c_{1} +1}\, \left (1+\sqrt {-{\mathrm e}^{8 x} c_{1} +1}\right )} \\ y &= -\frac {2 \,{\mathrm e}^{8 x} c_{1}}{\sqrt {-{\mathrm e}^{8 x} c_{1} +1}\, \left (-1+\sqrt {-{\mathrm e}^{8 x} c_{1} +1}\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 12.462 (sec). Leaf size: 102

DSolve[D[y[x],x]==y[x]*(2-y[x])*(4-y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {2 \left (-\sqrt {1+e^{8 (x+c_1)}}+e^{8 (x+c_1)}+1\right )}{1+e^{8 (x+c_1)}} \\ y(x)\to \frac {2 \left (\sqrt {1+e^{8 (x+c_1)}}+e^{8 (x+c_1)}+1\right )}{1+e^{8 (x+c_1)}} \\ y(x)\to 0 \\ y(x)\to 2 \\ y(x)\to 4 \\ \end{align*}