1-Compartment Model population question

[heyheyhey701]

A 1-compartment model can be used to study the population effects in the Great lakes. We make 2 assumptions:
- The polllutants are being added to a lake at a constant rate i, and that the pollutants and thoroughly mixed into the lake.
- The annual precipitation into the lake matches evaporation, so the flow rate F is also constant.

Let y(t) be the amount of pollutant in the lake and c(t) its concentration at time t.

The differential equation for y in this 1-compartment model is y' = - (f/v) y + i. Simply dividing this by V and recalling that c(t) = y(t)/v, we find that the differential for c is

c' = -(f/v)c + i/v

Solve the first order linear differential equation for the unknown function c(t) using the 7step method, treating F,V, and I as constants. Use the answer to compute teh long range concentration Cinfinity = lim as t approaches infinity of c(t)

 

[galactus]

I do not know what the 7-step method is, but your DE can be written as:

\(\displaystyle c'+\frac{f}{v}c=\frac{i}{v}\)

Find the integrating factor by looking at the coefficient of the linear term.

That is, \(\displaystyle e^{\int \frac{f}{v}dt}=e^{\frac{ft}{v}}\)

\(\displaystyle \frac{d}{dt}[ce^{\frac{ft}{v}}]=\frac{i}{v}e^{\frac{ft}{v}}\)

Now, integrate and finish up by solving for c in terms of t.

After solving for c in terms of t, you should be able to see the concentration as \(\displaystyle t\to \infty\)

Is this 7-step method the same as solving by finding the integrating factor?.

 

[Endatte]

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I do not know what the 7-step method is, but your DE can be written as:Find the integrating factor by looking at the coefficient of the linear term.That is, Now, integrate and finish up by solving for c in terms of t.After solving for c in terms of t, you should be able to see the concentration as Is this 7-step method the same as solving by finding the integrating factor?.