Help modeling differential equation.

[KFS]

The problem I'm stuck is 15. I added Example 4 so you can see what problem 15 is all about. What I've been trying is: the amount of pollutant in the first lake is y=(1-e^(-2.67*10^(-7)t)(2.51*10^6). So I multiplied y by the rate entering the second lake which is the one leaving the first lake. Then I multiply 10.67 by the amount of pollutant in the second lake which is what I have to find through a linear first-order differential equation. Then I have dc/dt=10.67y+10.67c. Something's wrong but I don't know what. I do not need help solving this kind of differential equations where I need help is FINDING THE DIFFERENTIAL EQUATION TO SOLVE.  

[studiot]

The amount of polutant entering the first lake per second is constant.

So the concentration of the pollutant gradually rises in the lake.

So the amount of pollutant leaving the first lake lake gradually rises with time.

So the amount of pollutant entering the second lake also rises with time.

So the equations of concentration are not the same for both lakes.

[KFS]

25 minutes ago, studiot said:

The amount of polutant entering the first lake per second is constant.

So the concentration of the pollutant gradually rises in the lake.

So the amount of pollutant leaving the first lake lake gradually rises with time.

So the amount of pollutant entering the second lake also rises with time.

So the equations of concentration are not the same for both lakes.

Thanks for answer but how do I represent the amount of pollutant in the second lake as a differential equation? I understood the process in Example 4 but not with the information of both lakes in problem 15.

10 minutes ago, KFS said:

Thanks for answer but how do I represent the amount of pollutant in the second lake as a differential equation? I understood the process in Example 4 but not with the information of both lakes in problem 15.

I failing at filling the gaps in the formula dy/dt= amount entering - amount leaving. How do I find the amount entering and leaving?

 

[joigus]

3 hours ago, studiot said:

The amount of polutant entering the first lake per second is constant.

So the concentration of the pollutant gradually rises in the lake.

So the amount of pollutant leaving the first lake lake gradually rises with time.

So the amount of pollutant entering the second lake also rises with time.

So the equations of concentration are not the same for both lakes.

 

3 hours ago, KFS said:

how do I represent the amount of pollutant in the second lake as a differential equation?

"Same" problem as first lake, but with time-depending input. Now take a look at the terms I've highlighted in Studiot's answer. You have a constraint.

[studiot]

13 minutes ago, joigus said:

You have a constraint.

I'd call it a helpful connection between the two equations.

🙂

If x is the quantity in a lake at time t then the change of x, dx/dt = rate of input - rate of output - both expressed as functions of the quantity x with time.

So the output of lake 1 (as the given expression) is the input to lake 2.

As Joigus says the ouput from lake 2 is formed in the same way.

Putting these two expressions into my little equation above gives you teh differential equation you seek.

 

You may need an integrating factor for this one , have you done these ?

 

[joigus]

2 minutes ago, studiot said:

I'd call it a helpful connection between the two equations.

I thought of saying "you have a helpful connection between the two equations."

I don't know what got into me. 

[KFS]

So dx/dt=2.67×10^(-7)y-10.67x, where y=(1-exp(-2.67×10^(-7)t)(2.51×10^6), is that correct?

Or dx/dt=2.67×10^(-7)y-1,067×10^(-6)x?

6 minutes ago, studiot said:

I'd call it a helpful connection between the two equations.

🙂

If x is the quantity in a lake at time t then the change of x, dx/dt = rate of input - rate of output - both expressed as functions of the quantity x with time.

So the output of lake 1 (as the given expression) is the input to lake 2.

As Joigus says the ouput from lake 2 is formed in the same way.

Putting these two expressions into my little equation above gives you teh differential equation you seek.

 

You may need an integrating factor for this one , have you done these ?

 

Yes I have done integrating factors. I have no problem solving these equations only setting up the problem.