Choose the correct graph of solution curve from the following options:

Question

Use the qualitative theory of Autonomous differential equations to sketch the graph of the solution of

$ y'=y(y-7)(y+7) , \\ y(0)=-9, \ y(0)=-1, \ y(0)=6, \ y(0)=8 \ $

Choose the correct graph of solution curve from the following options:

Answer:

The differential equation is

$ y'=y(y-7)(y+7) \ $

The equilibrium points are given by

$ y=0, \ 7 , \ -7 \ $

Also,

$ y'>0 \ $ on the interval $ \ (7,\infty) \ $

$ y'<0 \ $ on the interval $ \ (0,7) \ $

$ y'>0 \ $ on the interval $ \ (-7,0) \ $

$ y'<0 \ $ on the interval $ \ (-\infty,-7) \ $

Thus option $ \ (A) \ $ and $ \ (B) \ $ seems to be correct.

But which one should be answer?

I think the initial conditions need to apply now.

But I am unable to do it.

Help me finishing the answer.

Answer 1

The only correct solution is $A.$

$B$ is decreasing on $(-7,0)$ which is contradictory to the given equation.

Answer 2

It should be option (A). In (B) the graph is increasing for $(0,7)$ which contradicts your observation.