Chapter 7, Problem 1RE

a.

To determine

To sketch : the graph of the solutions that satisfy the initial conditions

1. $y\left(0\right)=-0.3$
2. $y\left(0\right)=1$
3. $y\left(0\right)=3$
4. $y\left(0\right)=4.3$

Explanation of Solution

Given information : The question is

A direction field for the differential equation is $y\text{'}=y\left(y-2\right)\left(y-4\right)$

Graph: To sketch the graph of the solutions that satisfy the initial condition

  $y\left(0\right)=-0.3$

  

  $y\left(0\right)=1$

  

  $y\left(0\right)=3$

  

  $y\left(0\right)=4.3$

  

Interpretation :

In the first case the purple curve represent the solution with initial condition $y\left(0\right)=-0.3$

In the second case the purple curve represent the solution with initial condition $y\left(0\right)=1$

In the third case the purple curve represent the solution with initial condition $y\left(0\right)=3$

In the fourth case the purple curve represent the solution with initial condition $y\left(0\right)=4.3$

b.

To determine

To calculate : the equilibrium solution

Answer to Problem 1RE

The equilibrium solution is $y=0,y=2,y=4$

Explanation of Solution

Given information : The question is

If the initial condition is $y\left(0\right)=c$ for what value of $c$

  $\underset{t\to \infty }{\lim }y\left(t\right)$ is finite

Calculation :

The slope field graph if $0\le c\le 4$ then $\underset{t\to \infty }{\lim }y\left(t\right)$ will approach the $0,2\&4$ or more

If $c=0$ then $\underset{t\to \infty }{\lim }y\left(t\right)=0$

If $0\le c\le 4$ then $\underset{t\to \infty }{\lim }y\left(t\right)=2$

If $c=4$ then $\underset{t\to \infty }{\lim }y\left(t\right)=4$

If $c>4$ then $\underset{t\to \infty }{\lim }y\left(t\right)=\infty$

If $c<0$ then $\underset{t\to \infty }{\lim }y\left(t\right)=-\infty$

  $y\text{'}=y\left(y-2\right)\left(y-4\right)$

The equilibrium solutions are where $y\text{'}=0$

So that would be when $y=0,y=2,y=4$