Math modelling diffrential equation question?

In summary, the equation x=Pqt changes when the amount of product in the container changes, but the new equation can be found by solving for P in the equation x=Pqt.
  • #1
Eveflutter
14
0
So there's this question I've to do. I got through until a certain point and now I'm stuck. >->

1&2The question:
The rate of deterioration of a product in a container is proportional to the amount of product present. At time t, the amount of product is x.
(i) State the diffrential equation relating x and t and solve the general solution to show that x=Ae-kt where A is an arbitrary contant and k is the contant of proportionality.
(ii) Before t=0, no product was present in the container. At t=0, P amount of the product was added to the container. When t=1, the amount of product in the container was Pq, where q is a constant such that 0<q<1. Show that x=Pqt.
(iii) When t=1 and again when t=2, another P amount of x was added to the container. Show that the amount of product in the container immediately after t=2 was P(q2+q+1)

3I got through with the first two parts:
(i) dx/dt=-kx
-> x=e-kt+c
-> x=Ae-kt , A=ec
(ii)when t=0, x=P
-> A=P
-> x= Pe-kt
when t=1, x=Pq
-> Pq=Pe-kt
-> q=e-kt
-> x=Pqt

I'm stuck at part (iii)
I tried reasoning it out but I don't know. Would the equation x=Pqt change since the intials are now changing twice (at t=1 and t=2)? But if so, how would I get the new ones from tha information?

Alright so as t approahes 1, x tends to Pq. At t=1 the amount of product changes to Pq+P. Then as t approaches 2, x deteriorates from Pq+P. But by how much?? Let's say it deteriorates till an amount B? Now at t=2, the amount of product goes to B+P.
I guess the P part of the required result (P(q2+q+1)) is that P in B+P. How do i get B?

I would really appreciate any help on this question, thank you!
 
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  • #2
Eveflutter said:
So there's this question I've to do. I got through until a certain point and now I'm stuck. >->

1&2The question:
The rate of deterioration of a product in a container is proportional to the amount of product present. At time t, the amount of product is x.
(i) State the diffrential equation relating x and t and solve the general solution to show that x=Ae-kt where A is an arbitrary contant and k is the contant of proportionality.
(ii) Before t=0, no product was present in the container. At t=0, P amount of the product was added to the container. When t=1, the amount of product in the container was Pq, where q is a constant such that 0<q<1. Show that x=Pqt.
(iii) When t=1 and again when t=2, another P amount of x was added to the container. Show that the amount of product in the container immediately after t=2 was P(q2+q+1)

3I got through with the first two parts:
(i) dx/dt=-kx
-> x=e-kt+c
-> x=Ae-kt , A=ec
(ii)when t=0, x=P
-> A=P
-> x= Pe-kt
when t=1, x=Pq
-> Pq=Pe-kt
-> q=e-kt
-> x=Pqt

I'm stuck at part (iii)
I tried reasoning it out but I don't know. Would the equation x=Pqt change since the intials are now changing twice (at t=1 and t=2)? But if so, how would I get the new ones from tha information?

Alright so as t approahes 1, x tends to Pq. At t=1 the amount of product changes to Pq+P. Then as t approaches 2, x deteriorates from Pq+P. But by how much?? Let's say it deteriorates till an amount B? Now at t=2, the amount of product goes to B+P.
I guess the P part of the required result (P(q2+q+1)) is that P in B+P. How do i get B?

I would really appreciate any help on this question, thank you!

Think of it this way. The amount ##P## at ##t=0## decreases according to ##Pq^t##. The amount ##P## added at ##t=1## decreases like ##Pq^{t-1}## (it's just 1 second delayed from the first). Then at ##t=2## you add ##Pq^{t-2}##. So the total after ##t=2## is ##Pq^t+Pq^{t-1}+Pq^{t-2}##. Correct?
 
  • #3
Dick said:
Think of it this way. The amount ##P## at ##t=0## decreases according to ##Pq^t##. The amount ##P## added at ##t=1## decreases like ##Pq^{t-1}## (it's just 1 second delayed from the first). Then at ##t=2## you add ##Pq^{t-2}##. So the total after ##t=2## is ##Pq^t+Pq^{t-1}+Pq^{t-2}##. Correct?
Ooooh! I understood what you were doing but not why you were doing it but it took me a couple minutes to completely get it. Thank you so much! I really appreicate your answer :)
 

Related to Math modelling diffrential equation question?

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It represents the relationship between a function and its rate of change, and is commonly used to model various natural phenomena in science and engineering.

2. How is differential equation used in math modelling?

Differential equations are used in math modelling to describe the behavior of systems that change over time. These equations can help predict future outcomes and understand the underlying processes of complex systems.

3. What is the difference between ordinary and partial differential equations?

Ordinary differential equations involve a single independent variable, while partial differential equations involve multiple independent variables. Ordinary differential equations are typically used to model systems with one variable changing over time, while partial differential equations are used to model systems with multiple variables changing over time and space.

4. Can all real-world phenomena be modeled using differential equations?

No, not all real-world phenomena can be accurately modeled using differential equations. Some systems may be too complex to be described by a single equation, or may involve random or chaotic factors that cannot be captured by mathematical models.

5. What are the main methods for solving differential equations?

There are several methods for solving differential equations, including separation of variables, substitution, and using numerical methods such as Euler's method or Runge-Kutta methods. The appropriate method will depend on the specific form and complexity of the equation.

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