I need to find the partial derivative
$\frac{\partial}{\partial t} \int_{0}^{\infty}e^{-a\tau}I(x,t-\tau)\ d\tau$.
The answer is given by
$I(x,t)-a\int_{0}^{\infty}e^{-a\tau}I(x,t-\tau)\ d\tau$.
What rule should I use to get the above answer?
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It is just a variable stands for infected cells concentration. – Fatimah Aug 21 '19 at 16:04
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1Your comment does not say what the meaning of $I(x,t)$ is. – callculus42 Aug 21 '19 at 16:06
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It represents the concentration of cells which depends on space and time. – Fatimah Aug 21 '19 at 16:13
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2@Fatimah We got that. That doesn't tell us any of its mathematical properties. Does it, for instance, have a functional expression? Or does it fulfill a certain differential equation? Or anything else like that? – Arthur Aug 21 '19 at 16:20
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Yes it fulfills a certain delay partial differential equation within a system with distributed delay. – Fatimah Aug 21 '19 at 16:25
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Well, then maybe that is essential to solving your problem. I suggest you include it in your post. – Arthur Aug 21 '19 at 16:48
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No sir the solution doesn’t depend on the equation of $I$. It can be done by inserting the partial derivative inside the integral, using the fact that $\frac{\partial}{\partial t}=-\frac{\partial}{\partial \tau}$, and finally using integration by parts. I have just realized the answer. Thanks for you all. – Fatimah Aug 21 '19 at 18:34