Modeling differential equationI am trying to model some differential equation which can explain buying some assets that can increase itself like a virus.So, let's assume my money is y(t), and number of assets is n(t).- y(t) to a certain level c, it will increase the n(t) by +1- n(t) generate profit r %so I can see something like d y ( t ) / d t = y 0 + r ⋅ n ( t ) d n ( t ) / d t = I f ( y ( t ) ≥ c , y ( t ) − 20 a n d n + 1 , n )However, I have no clue how I can model this discrete-continuous variable differential equation. Is there any way to model the system and get the solution?

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Modeling differential equationI am trying to model some differential equation which can explain buying some assets that can increase itself like a virus.So, let&#039;s assume my money is y(t), and number of assets is n(t).- y(t) to a certain level c, it will increase the n(t) by +1- n(t) generate profit r %so I can see something like  d  y  (  t  )      /    d  t  =      y    0    +  r  ⋅  n  (  t  )    d  n  (  t  )      /    d  t  =  I  f     (  y  (  t  )  ≥  c  ,  y  (  t  )  −  20     a  n  d     n  +  1  ,  n  )However, I have no clue how I can model this discrete-continuous variable differential equation. Is there any way to model the system and get the solution?

Lillianna Salazar · Accepted Answer

Step 1With the information you have given, one can make a set of recurrence relations as follows:      {                                        y                          k              +              1                                =                      y                          k                                +          r                      n            k                    −          20          I          (                      y            k                    ≥          c          )                                                  n                          k              +              1                                =                      n            k                    +          I          (                      y            k                    ≥          c          )                        If you want to have a continuous evolution of the growth part, then you&#039;ll have to break up the equation as soon as y reaches c, you reset the system and put the initial condition   y  =  c  −  20. You&#039;ll have something like            d      y              d      t        =  r  n  (  t  )   with   y  &amp;lt;  cand when  y  (  t  )  ≥  c    ⟹    y  (  t  +  d  t  )  =  y  (  t  )  −  20   and   n  (  t  +  d  t  )  =  n  (  t  )  +  1    .Where by the use of dt I mean to represent an infinitesimal time jump. But what you&#039;ll have is a continuous evolution of the equation up until y reaches the limit c, at that point a reset in your initial conditions of the differential equation and you let the equation evolve continuously again. In other words, the result will be a stepwise continuous function. It will consist of straight rising segments from left to right, topped off at c, then falling back to   c  −  20 and rising again. The speed at which the saw tooth will develop will increase, each successive saw tooth having a length of 20/rn time units.

Modeling differential equation. So, let's assume my money is y(t), and number of assets is n(t). - y(t) to a certain level c, it will increase the n(t) by +1, n(t) generate profit r %

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