# Integrating Factor Question

#### themurgesh

##### New member
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
Re: Differential equation problem

can you post the question ?

#### themurgesh

##### New member
Re: Differential equation problem

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

#### Amer

##### Active member
Re: Differential equation problem

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

#### Ackbach

##### Indicium Physicus
Staff member
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 $$e^{(\lambda_3+\mu_3)t}$$ 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?

#### themurgesh

##### New member
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

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