Solving for P02 using Integrating Factor?

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Discussion Overview

The discussion revolves around solving a differential equation using the integrating factor method. Participants are examining the relationships between various functions, specifically \(P_{01}(t)\) and \(P_{02}(t)\), and how to derive one from the other using integration techniques. The focus is on the mathematical manipulation of equations and the application of integrating factors.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • The original differential equation is stated as \(P'_{02}(t)+(\lambda_3+\mu_3)P_{02}(t)=P_{01}(t) \lambda_2\).
  • One participant expresses confusion about how to derive \(P_{02}(t)\) from the earlier equations and the lengthy expression provided in the referenced article.
  • Another participant questions why the term \(e^{(\lambda_3+\mu_3)t}\) was not canceled from the first equation, suggesting that differentiation must occur before cancellation.
  • There is a request for clarification on the definitions of \(P_{01}\) and \(P_{02}\) from the original poster.
  • Participants are discussing the implications of the derivative operator in the context of the equations presented.

Areas of Agreement / Disagreement

The discussion contains multiple viewpoints regarding the manipulation of the equations and the use of integrating factors. There is no consensus on how to proceed with deriving \(P_{02}(t)\) from the given equations, and participants are seeking further clarification and information.

Contextual Notes

Participants have not fully defined the terms \(P_{01}\) and \(P_{02}\), and the discussion lacks clarity on the assumptions underlying the equations. The original differential equation and the steps leading to the current expressions are not fully resolved.

themurgesh
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Hello,


I am solving an equation using integrating factor. I have come up to a specific point which is $$\dfrac{d}{dt} P_{02}(t) \cdot e^{(\lambda_3+\mu_3)t}=\lambda_2 \cdot P_{01}(t) \cdot e^{(\lambda_3+\mu_3)t}$$

from the previous equation, I have found $$P_{01}(t)=\lambda_1 \int_0^t e^{-(\lambda_1-\mu_1)s}\;e^{-(\lambda_2+\mu_2)(t-s)} ds $$

I have used both these facts to come to an answer which is a very lengthy term. However, the article I am reading also states that

$$P_{02}(t)=\lambda_1 \int_0^t e^{-\lambda_1 s}e^{-\mu_1 s} \int_0^{t-s}\lambda_2 e^{-\lambda_2 q} e^{-\mu_2 q} e^{-(\mu_3+\lambda_3)(t-s-q)}\;dq\;ds$$

I am getting confused about how do I use the first two equations to achieve the third one? I would appreciate any suggestions. Thank You.

 
Last edited:
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Re: Differential equation problem

can you post the question ?
 
Re: Differential equation problem

sorry.. i had some latex errors.. i hope you can see the question now.
 
Re: Differential equation problem

why you did not cancel the term e^{(\lambda_3+\mu_3)t} from the first equation
and what is P_{01},P_{02} ?
 
Re: Differential equation problem

themurgesh said:
Hello,


I am solving an equation using integrating factor. I have come up to a specific point which is $$\dfrac{d}{dt} P_{02}(t) \cdot e^{(\lambda_3+\mu_3)t}=\lambda_2 \cdot P_{01}(t) \cdot e^{(\lambda_3+\mu_3)t}$$

from the previous equation, I have found $$P_{01}(t)=\lambda_1 \int_0^t e^{-(\lambda_1-\mu_1)s}\;e^{-(\lambda_2+\mu_2)(t-s)} ds $$

I have used both these facts to come to an answer which is a very lengthy term. However, the article I am reading also states that

$$P_{02}(t)=\lambda_1 \int_0^t e^{-\lambda_1 s}e^{-\mu_1 s} \int_0^{t-s}\lambda_2 e^{-\lambda_2 q} e^{-\mu_2 q} e^{-(\mu_3+\lambda_3)(t-s-q)}\;dq\;ds$$

I am getting confused about how do I use the first two equations to achieve the third one? I would appreciate any suggestions. Thank You.

Amer said:
why you did not cancel the term e^{(\lambda_3+\mu_3)t} from the first equation
and what is P_{01},P_{02} ?

If the derivative operator is acting on that term, you'd have to differentiate before canceling.

To the OP'er: could you please post the original DE?
 
Re: Differential equation problem

Ackbach said:
If the derivative operator is acting on that term, you'd have to differentiate before canceling.

To the OP'er: could you please post the original DE?

Sorry. I should have posted the original DE before

$$P'_{02}(t)+(\lambda_3+\mu_3)P_{02}(t)=P_{01}(t) \lambda_2$$


and then I used integrating factor $$e^{\int_0^t (\lambda_3+\mu_3) dt}$$ and arrive at the equation

$$\dfrac{d}{dt} [P_{02}(t) \cdot e^{(\lambda_3+\mu_3)t}]=\lambda_2 \cdot P_{01}(t) \cdot e^{(\lambda_3+\mu_3)t}$$
 

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