Differential equation: the law of natural growth and the law of natural decayI understand that d y d x = k ∗ y and when k&gt;0 this is the law of natural growth and when k&lt;0 this is the law of natural decay, but my textbook gives an example of radioactive decay as follows which confuses me:Radioactive substances decay by spontaneously emitting radiation. If is the mass remaining from an initial mass of the substance after time t, then the relative decay rate − 1 m d m d t has been found experimentally to be constant. (Since d m d t is negative, the relative decay rate is positive.) It follows that: d m d t = k mwhere k is a negative constant. In other words, radioactive substances decay at a rate proportional to the remaining mass. This means that we can use to show that the mass decays exponentially: m ( t ) = m ( 0 ) e k t (This equation I understand and accept, the above two confuses me)Now eventually when the example becomes numerical, then of course k becomes a negative number since it is natural decay. However, the above general notation explanation confuses me because it keeps changing the sign of k=relative decay rate(from negative(1) to positive(2), but eventually when numerically worked it turns out to be a negative constant since it's natural decay). I know that k must be less than zero for decay but I'm just trying to fully grasp the notation signs that the textbook uses in the explanation above.

Question

Differential equation: the law of natural growth and the law of natural decayI understand that             d      y              d      x        =  k  ∗  y and when k&amp;gt;0 this is the law of natural growth and when k&amp;lt;0 this is the law of natural decay, but my textbook gives an example of radioactive decay as follows which confuses me:Radioactive substances decay by spontaneously emitting radiation. If is the mass remaining from an initial mass of the substance after time t, then the relative decay rate            −      1        m              d      m              d      t      has been found experimentally to be constant. (Since             d      m              d      t       is negative, the relative decay rate is positive.) It follows that:            d      m              d      t        =  k  mwhere k is a negative constant. In other words, radioactive substances decay at a rate proportional to the remaining mass. This means that we can use to show that the mass decays exponentially:  m  (  t  )  =  m  (  0  )      e          k      t       (This equation I understand and accept, the above two confuses me)Now eventually when the example becomes numerical, then of course k becomes a negative number since it is natural decay. However, the above general notation explanation confuses me because it keeps changing the sign of k=relative decay rate(from negative(1) to positive(2), but eventually when numerically worked it turns out to be a negative constant since it&#039;s natural decay). I know that k must be less than zero for decay but I&#039;m just trying to fully grasp the notation signs that the textbook uses in the explanation above.

cercimw · Accepted Answer

On one hand you can say k is defined positive. Then -k is negative. The differential equation in case of decay is then             d      m              d      t        =  −  k  m. The solution then is   m  =  C  ⋅      e          −      k      t      On the other hand you can say that k can be positive or negative. Then the differential eqution is             d      m              d      t        =  k  mThe solution then is   m  =  C  ⋅      e          k      t      If you have an exponential decay k gets negative.In both cases you get a negative expoenent, if you have an exponential decay.

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