What is the proper DE for those questions?

Question

A tank starts with 500 liters of water with 1 kg of salt dissolved in it. A salt and water mixture with concentration 0.1 kg/L is poured into the tank at a rate of 2 L/min. The mixture is drained at 4 L/min. Assuming that the mixture is well stirred (uniformly distributed) (a) write a DE together with an initial condition that describes the amount of salt m(t) in the tank at time t.

I tried to answer this question but got stuck on getting the right DE, which is the first step. I got (dm/dt) = 0.2 - ((1/125)-(2/t)m) As I proceeded, I found out that I was mistaken. Could someone correct me please? I just need the to know the right DE.

Answer 1

There is 2L of water being added and 4L being drained every minute. So the total amount of water is $w(t) = 500 - 2t$.

Salt is being added at 0.2 kg/m, and being drained at 4 times the proportion of salt in the water. So your ODE should be

$$\frac{dm}{dt} = 0.2 - 4\frac{m(t)}{w(t)} = 0.2 - \frac{4 m(t)}{500 - 2t}$$

From here you need the integrating factor method.

Answer 2

HINT:

If 1 and 2 suffixes refer to pour-in and pour-out situations,

$$ m = m1 - m2 $$

$$ \frac{dm}{dt} = \frac{dm1}{dt} - \frac{dm2}{dt} $$

$$ \frac{dm}{dt} = \frac{dm1}{dV1}\cdot \frac{dV1}{dt}- \frac{dm2}{dV2} \cdot \frac{dV2}{dt} = (0.1) (2) -\frac{m1-m}{dV2}\cdot 4 $$