Explanation of proof nedeed: Why is y ′ = c ⋅ y always a exponential growth/decay function?I know, that a solution of y ′ = c ⋅ y is y = a ⋅ e c t and it's clear how to calculate this.I want to proof, that all solutions of a function describing a change of population, that is proprotional to the population, like y ′ = c ⋅ y is a function of exponential growth (or decay.I found a proof:Let g be an other solution, whilst g is not describing an exponential growth or decay. We show ( g e c t ) ′ = 0 It's fine to me how they show it's zero. But where does ( g e c t ) ′ come from and why are they using it here, what does it mean?

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Explanation of proof nedeed: Why is        y    ′    =  c  ⋅  y  always a exponential growth/decay function?I know, that a solution of       y    ′    =  c  ⋅  y is   y  =  a  ⋅      e          c      t       and it&#039;s clear how to calculate this.I want to proof, that all solutions of a function describing a change of population, that is proprotional to the population, like       y    ′    =  c  ⋅  y is a function of exponential growth (or decay.I found a proof:Let g be an other solution, whilst g is not describing an exponential growth or decay. We show   (      g          e              c        t                  )    ′    =  0 It&#039;s fine to me how they show it&#039;s zero. But where does   (      g          e              c        t                  )    ′   come from and why are they using it here, what does it mean?

emarisidie6 · Accepted Answer

It&#039;s fine to me how they show it&#039;s zero. But where does   (      g          e              c        t                  )    ′   come from and why are they using it here, what does it mean?I&#039;m not sure what it   g  (  t  )      /        e          c      t       means in and of itself. But it a means to an end. (excuse my play on words there).Our aim is to reveal that a feature satisfying a sure differential equation must take a sure form. The principal equation of this form, and its solution, is the reality that a characteristic whose by-product is zero (on an c language) need to be a consistent (on that interval).So we prepare dinner up a associated function which, if it have been consistent, would tell us the feature we&#039;re originally looking at ought to have the shape we are claiming it to have. Then we take the derivative of the associated function and display it&#039;s miles zero.Because we want to show   g  (  t  )  =  a      e          c      t       for some a, we construct the quotient   h  (  t  )  =  g  (  t  )      /        e          c      t      . This function is constant if and only if g has the form we claim it does. Then we prove that h is constant by showing       h    ′    (  t  )  =  0.This is not obvious to anyone seeing it for the first time, but now you know the technique: find something you want to be constant, and prove that it must be constant by showing its derivative is zero.

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