# Differential Equations Tank problem

1. Feb 5, 2010

### SpiffyEh

1. The problem statement, all variables and given/known data
The problem states that at time 0 the tank has 10 lb of salt dissolved in 100 gallons of water. The capacity is 200. Assume that water containing 1/8 lb of salt per gallon is entering the tank at a rate of 2 gal/min and the mizture is draining from the tank at a rate of 1 gal/min.
a) set up the initial value problem
b)solve using method of integrating factors.

2. Relevant equations
t:time
y: amount of salt in tank (lbs)
v:volume of water (lbs)
v(t) = 100+t

3. The attempt at a solution

a) for the equation dy/dt = 1/4 - y/(100-t) y(0) = 10
b) *this is where i think i'm messing up...

dy/dt +(1/(100-t))y = 1/4

$$\mu$$(t) = $$e^{\int\frac{1}{100-t}dt}$$

$$e^{-ln|100-t|}$$ = $$e^{ln|100-t|^{-1}}$$ = $$(100-t)^{-1}$$

= $$\frac{1}{100-t}$$

so i multiply through and get

($$\frac{1}{100-t}$$ * y)' = $$\frac{1}{4(100-t)}$$
$$\frac{1}{100-t}$$ * y = $$\int$$$$\frac{1}{4(100-t)}$$dt
$$\frac{1}{100-t}$$ * y = ln|4(100-t)| +C
then solving for y
y = (100-t)*(ln|4(100-t)| +C)

This doesn't seem right to me for some reason, I just want to make sure i'm doing this problem right. Can someone please let me know? Thank you.

2. Feb 5, 2010

### tiny-tim

Hi SpiffyEh!
Technique looks fine, but

i] shouldn't it be 100+t ?

ii] your 4 should be outside the ln

3. Feb 5, 2010

### SpiffyEh

oops, sorry yeah it should be plus t. Should it be outside the ln? or should i just multiply the values by 4?

4. Feb 6, 2010

### tiny-tim

(just got up :zzz: …)

Not following you … aren't they the same?

5. Feb 6, 2010

### SpiffyEh

yeah, i just thought it would be easier to multiply it through

6. Feb 6, 2010

### SpiffyEh

with the initital condition i got c = 1/10 - ln400
so.. the whole equation is....
y(t) = (100+t)*(ln|400+4t|+1/10-ln400)

does that look right?

Last edited: Feb 6, 2010
7. Feb 6, 2010

### tiny-tim

Nooo

your integrating factor was for 100 - t.

Now it's 100 + t, the integrating factor is completely different.

8. Feb 6, 2010

### SpiffyEh

it was supposed to be 100+t, i accidently changed it in the middle of all the equations