Welcome to our community

Be a part of something great, join today!

Integrating Factor Question

themurgesh

New member
Mar 29, 2012
4
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:

Amer

Active member
Mar 1, 2012
275
Re: Differential equation problem

can you post the question ?
 

themurgesh

New member
Mar 29, 2012
4
Re: Differential equation problem

sorry.. i had some latex errors.. i hope you can see the question now.
 

Amer

Active member
Mar 1, 2012
275
Re: Differential equation problem

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

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,192
Re: Differential equation problem

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.

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

themurgesh

New member
Mar 29, 2012
4
Re: Differential equation problem

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

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