# Homework Help: Mixing Problem (variable volume)

1. Mar 17, 2008

### Tyst

1. The problem statement, all variables and given/known data
A tank contains 150L of water in which initially 30g or salt is dissolved. Brine runs in at a rate of 4 L/m and each litre contains 1.5g of dissolved salt. The mix in the tank is kept uniform. Brine runs out at 3L/m. What is that amount of salt in the tank at any time t?
Note that the volume of brime in the tank is not constant in time

2. Relevant equations

$$y' +p(x)y = r(x)$$

General solution to first order, linear ODE

$$y(t) = exp(-\intp(p(t)) dt) [\int r(t)\exp(\intp(p(t))dt) dt +C]$$

3. The attempt at a solution

Input/Output Gives

$$y(t)=(6-3y)/(150+t)$$
-
$$y'+3y = 6/(150+t)$$

Substituting these values into the general equation gives me

y(t) = e^(-3t)[\nt(6/(150+t)*e^(3t) dt) + C]

And from here ... well... some sort of a clue as to where to go next would be very nice.
Basically i am having trouble with that integral, is it possible to take $$\int(ln((6/(150+t)*\exp^(3t) dt)$$ somehow? I'm not sure if/how to make this step.
I can take the log of the other parts of the equation...

ln(y(t)) = ln(exp^3t)+ [this is the bit i'm not sure of] + ln(C)

Last edited: Mar 17, 2008
2. Mar 17, 2008

### rocomath

3. Mar 17, 2008

### Tyst

Yes, thanks for showing me that - perhaps it's just the tiredness setting in, or perhaps i'm just completely dense! But i am still not able to work this out.

4. Mar 17, 2008

### Tyst

Well, i'm an idiot :P solved it now! Thanks!