Hello!
I have been given an equation as following;
where
and are positive constants
1)
I am then to find the constants and so the equation can be given as this:
2)
And at last find a general solution when
Don't know where to start actually...
Any help is greatly appreciated!
This is the logistic equation, correct?. Here is a general tutorial to discuss the solutions of the logistic equation.
Let's start from scratch. Okey-doke.
for y>0. A and a are constants.....[1]
Separate variables:
Integrate:
.....[2]
Where is a constant of integration. By use of partial
fractions as Calculus26 showed:
Therefore, [2] becomes:
or
when we integrate.
Therefore, .
Exponentiating gives:
where is now our arbitrary constant.
Since y>0 by our assumption, if we also assume y<A, the absolute
value signs are done with.
We can also consider either case by simply letting the arbitrary cnstant C
be positive or negative.
Therefore, .
Solving for y:
.....[3]
which is the general solution of [1].
To evaluate C we need the initial conditions:
Sub into [3]:
Hence, thus, and therefore,
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Now, you asked what happens to y when t-->infinity.
Suppose a>0 and A>0. If y(t) represents population then certain models
say that
If we have a small population(small y), then
and the birth rate exceeds the death rate. Thus, the population is growing.
We take note that:
Since
This is a curious result since it says the population approaches a limiting
value A independent of the initial population size we call
A is called the limiting population that the syatem can support. If
, the population decreases.
\
If the population increases.
We can also take note that when y=A, [1] becomes y'=0 and we are at
equilibrium(birth rate=death rate).
Here is a haphazard 'paint' graph and I hope this little tutorial helps some to understand it.