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#### themurgesh

##### New member

- Mar 29, 2012

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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.

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.

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