Easy to visualize example:
You have a 5 gallon bucket completely full of sea (~3.5% salt) water. To avoid unit conversions, lets say that is 3.5% salt by volume instead of weight. You take a garden hose, flowing fresh (no salt) water at one gallon/minute and run it into the bucket, allowing it overflow. You have your pet monkey stir it constantly with a stick, so that all the water in the bucket is at the same salt concentration. *
Now it should be obvious that if you wait a VERY long time, the water in bucket will have virtually no salt. But what how much salt is in the bucket after 1 minute, 2 minutes, 2.3 minutes, etcetera?
To solve this problem, consider that the 1/5 of the salt in the bucket is overflowing each minute. That is to say that the rate that salt is going out of the bucket depends on how much is there at the time. If we call the amount of salt in the bucket at any given time "x(t) " , The differential equation that discribes this is:
dx(t)/dt=-((1gal/minute)/5gal) * x(t) = -x(t)/5 per minute. (eq 1)
Stated in English this is approximatly “Each minute we lose 1/5 of whatever our current amount of salt was”. This ignores the fact that amount of salt is NOT the same during each minute, it is constantly decreasing. Eq 1 accounts for this, while the version in english does not.
A complete solution to a differential equation requires an initial condition, and in this case that is:
x(0) = 3.5% * 5 = 0.175 gal salt. (eq 2)
If we take the time integral of both sides of eq1, we get:
x(t)= -1/5* ( the integral of x(t)) . Since we have studied calculus, we know that it is only the exponential function (e**x) that is it’s own integral. (this is where “the best way to solve a DE is to already know the answer” comes in) so we get:
x(t)=x(0) * (e**(-t/5))
(sorry for the lack of intermediate steps…I don’t know how to show integrals)
The final solution to our problem is:
x(t)=.175 * e**(-t/5)
X(1)=.143 gallons salt
x(2)=.117 gal salt,
x(2.3)=.110 gal salt, etc.
After a thousand minutes the monkey will become bored and run off, and the amount of salt is:
X(1000)= 242 * 10**-90 gal.
OR pretty much no salt at all unless you subscribe to the theories of homeopathy.
- I don’t know why you would do this either…probably the monkey’s idea. DEs are also a nice way to express radioactive decay.