What exactly does determine the behavior mean (differential equations)

[dwilmer]

What exactly does "determine the behavior" mean (differential equations)

Homework Statement

Draw a direction field for the given differential equation. Based on the diection field, determine the behavior of y as t approaches infinity. If this behavior depends on the initial value of y at t=0. describe the dependency.

y' = 3 + 2y

Homework Equations

1. How can I determine behavior of y with respect to t if t is not in the equation?

2. What exactly does it mean to ask "determine the behavior". Are there only a few scenarios for the behavior and if so what are they? The answer says that y diverges from (-3/2) as t approaches infinity, but I don't know (or forgot) what "diverges" means exactly.

3. Based on question 2 (above) does that mean that when a problem wants the behavior, it will either converge or diverge, and if so what is y (or t) converging/diverging onto?

The Attempt at a Solution

I understand that when i make RHS = zero, this is the equilibrium position. That is, y = -3/2, and values bigger than this slope will be pos.. values less and slope will be neg.
I also undherstand that y' is the slope, and that y' is same as saying dy/dt.
I also understand concept of the direction field.. It is (in my own words) lots of slope values for each value of t (or y, which is also confusing me...(theres no t in the original equation))

(PS: this is question #3, 1.1 in boyce and diprima 8th edition)
thanks for any help!

 

[Mark44]

Homework Statement

Draw a direction field for the given differential equation. Based on the diection field, determine the behavior of y as t approaches infinity. If this behavior depends on the initial value of y at t=0. describe the dependency.

y' = 3 + 2y

Homework Equations

1. How can I determine behavior of y with respect to t if t is not in the equation?

t is not explicitly in your differential equation, but both y and y' are functions of t. At each point (t, y) in the plane you can draw a short tangent line whose slope is 3 + 2y. It should be clear that along any horizontal line, the slopes of these tangent line segments will be the same.

2. What exactly does it mean to ask "determine the behavior". Are there only a few scenarios for the behavior and if so what are they? The answer says that y diverges from (-3/2) as t approaches infinity, but I don't know (or forgot) what "diverges" means exactly.

"Determine the behavior" in this context means to say what y will do as x gets very large or very negative. "Diverges" means separates from. If you draw the direction field you might get a sense of what the set of solutions to your diff. equation looks like.

3. Based on question 2 (above) does that mean that when a problem wants the behavior, it will either converge or diverge, and if so what is y (or t) converging/diverging onto?

The Attempt at a Solution

I understand that when i make RHS = zero, this is the equilibrium position. That is, y = -3/2, and values bigger than this slope will be pos.. values less and slope will be neg.
I also undherstand that y' is the slope, and that y' is same as saying dy/dt.
I also understand concept of the direction field.. It is (in my own words) lots of slope values for each value of t (or y, which is also confusing me...(theres no t in the original equation))

Lots of slope values, with one at each point (t, y).

(PS: this is question #3, 1.1 in boyce and diprima 8th edition)
thanks for any help!

 

[dwilmer]

thanks for help.
Could you clarify a little further, when you say:

"If you draw the direction field you might get a sense of what the set of solutions to your diff. equation looks like."

because I've already drawn the direction field, but what do you mean by "set of solutions"?? As far as I can tell the equation has a solution everywhere except when y=-3/2. So if there are that many solutions, then all i see is slopes increasining when larger than -3/2 and decreasing when less than -3/2, and when exactly -3/2, it is horizontal..
So when you say "set of solutions" is that every other point except for when y=-3/2?

 

[Mark44]

The set includes that one, too. You have a differential equation without an initial condition y(0). Each possible value of y(0) determines a specific solution.