Breakdown of a Logistic Equation

[defaultusername]

Homework Statement

I feel so stuck.
Given the Logistic Equation:
$$\frac{dP}{dt}=kP(1-\frac{P}{A})$$

a.). Find the equilibrium solutions by setting $$\frac{dP}{dt}=0$$ and solving for P.
b.). The equation is separable. Separate it and write the separated form of the equation.
c.). Use partial fraction decomposition and then integrate both sides of the equation to solve for P.

Homework Equations

$$\frac{dP}{dt}=kP(1-\frac{P}{A})$$

The Attempt at a Solution

a.) $$\frac{dP}{dt}=0$$
$$⇒P=0, A$$

b.) $$⇒\frac{1}{P}+(\frac{\frac{1}{A}}{1-\frac{P}{A}})dP=k dt$$
$$⇒(\frac{dP}{1-\frac{P}{A}})=k dt$$

c.) $$⇒(\frac{dP}{1-\frac{P}{A}})=k dt$$
$$⇒P=\frac{B}{P}+\frac{C}{1-\frac{P}{A}}$$

I am not even sure if I am doing everything correctly or not. I need to find a common denominator to solve for B and C but my attempts always end up as a huge mess.

 

[Mark44]

Homework Statement

I feel so stuck.
Given the Logistic Equation:
$$\frac{dP}{dt}=kP(1-\frac{P}{A})$$

a.). Find the equilibrium solutions by setting $$\frac{dP}{dt}=0$$ and solving for P.
b.). The equation is separable. Separate it and write the separated form of the equation.
c.). Use partial fraction decomposition and then integrate both sides of the equation to solve for P.

Homework Equations

$$\frac{dP}{dt}=kP(1-\frac{P}{A})$$

The Attempt at a Solution

a.) $$\frac{dP}{dt}=0$$
$$⇒P=0, A$$

b.) $$⇒\frac{1}{P}+(\frac{\frac{1}{A}}{1-\frac{P}{A}})dP=k dt$$

I can't tell what you did here (above). Start with the given diff. equation and separate it, using my hint below.
$$\frac{dP}{dt}=kP(1-\frac{P}{A})\\
\Rightarrow \frac{dP}{dt}=\frac k A P(A - P)$$
Now it should be easier to separate.

$$⇒(\frac{dP}{1-\frac{P}{A}})=k dt$$

c.) $$⇒(\frac{dP}{1-\frac{P}{A}})=k dt$$
$$⇒P=\frac{B}{P}+\frac{C}{1-\frac{P}{A}}$$

I am not even sure if I am doing everything correctly or not.

What you did is definitely wrong. You should get P as a function of t.

I need to find a common denominator to solve for B and C but my attempts always end up as a huge mess.

 

[epenguin]

Following where you write b) you have just made a mistake in getting the partial fractions.

In the following lines, you've lost yourself - you drop a term for no reason and integrate incorrectly.

You will need to integrate a LHS with respect to P, and right hand side with respect to t.

Revise the integrals of things like dP/P if you have forgotten.

 

[defaultusername]

Thanks for your help!

 

[epenguin]

Thanks for your help!

See my sig.