Expert: Paul Klarreich - 8/10/2009

Question
QUESTION: 1.what is exponential equations and why is it important? :)
2.how to show or prove that in y=y-knot e^(kt), y-knot is initial and k is the  rate?

ANSWER: Questioner: ain^^
Country: Malaysia
Category: Advanced Math
Private: No
Subject: exponential

Question: 1.what is exponential equations and why is it important? :)

-- see the (new) subject line for this question.

2.how to show or prove that in y=y-knot e^(kt), y-knot is initial and k is the  rate?

-- You solve the differential equation of growth and decay.



---------- FOLLOW-UP ----------

QUESTION: 1.what do you mean by see the subject line?
where is the subject line?
2.btw, by solving the equation of growth and decay i can really answered my question before?
its really confusing. dont you have any other way to prove that y-knot is initial and k is the rate? (in a proper and clearer way)
thanx. :)

Answer
Questioner: ain^^
Country: Philippines
Category: Advanced Math
Private: No
Subject: Exponential Decay.
Question: QUESTION: 1.what is exponential equations and why is it important? :)
2.how to show or prove that in y=y-knot e^(kt), y-knot is initial and k is the  rate?

ANSWER: Questioner: ain^^
Country: Malaysia
Category: Advanced Math
Private: No
Subject: exponential

Question: 1.what is exponential equations and why is it important? :)

-- see the (new) subject line for this question.

2.how to show or prove that in y=y-knot e^(kt), y-knot is initial and k is the  rate?

-- You solve the differential equation of growth and decay.



---------- FOLLOW-UP ----------

QUESTION: 1.what do you mean by see the subject line?
where is the subject line?

>> At the top.  It says "Exponential decay."
2.btw, by solving the equation of growth and decay i can really answered my question before?
its really confusing. dont you have any other way to prove that y-knot is initial and k is the rate? (in a proper and clearer way)
thanx. :)
..........................
You DEFINE y0 as the amount present at time  t = 0.  (That is what 'initial' means.)  Then you write the D.E. that states:

THE RATE OF DECAY IS PROPORTIONAL TO THE AMOUNT PRESENT.

THE RATE OF DECAY  means  dy/dt

IS PROPORTIONAL TO means  'equals  k times'

THE AMOUNT PRESENT  is  y.

dy/dt = ky

That is the equation.

Solve by separation of variables:

dy
-- = k dt
y

Integrate:

ln y = kt + c

y = C e^(kt),  where  C = e^c

Now at t = 0:

y0 = C, so the equation is:

y = y0 e^(kt)