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I want to differentiate the Volterra integral equation $\phi(t) + \int_0^t (t - \xi) \, \phi(\xi) \, \mathrm{d}{\xi} = \sin{2t}$.

Am I right in thinking that the integral can just be removed like so?

$\frac{\mathrm{d}}{\mathrm{d}t} \left( \phi(t) + \int_0^t (t - \xi) \, \phi(\xi) \, \mathrm{d}{\xi} = \sin{2t} \right)\\ = \phi'(t) + (t - \xi) \, \phi(\xi) = 2\cos{2t}$

And in general, how do you differentiate equations involving integral expressions? I haven't encountered equations of this nature before.

Harry Peter
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user125340
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1 Answers1

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I think you confused the limit of integration and its variable.

If we define $F:\mathbb{R}\rightarrow\mathbb{R}$ by $$F(t)=\int_0^t(t-\xi)\phi(\xi)d\xi=t\int_0^t\phi(\xi)d\xi-\int_0^t\xi\phi(\xi)d\xi$$ Then deriving both sides using linearity, the product rule and the fundamental theorem of calculus we get $$F'(t)=\frac{d}{dt}\left(t\int_0^t\phi(\xi)d\xi\right)-\frac{d}{dt}\left(\int_0^t\xi\phi(\xi)d\xi\right)$$ $$F'(t)=1\int_0^t\phi(\xi)d\xi+t\phi(t)-t\phi(t)=\int_0^t\phi(\xi)d\xi$$

In general you should separate the "$t$"s and "$\xi$"s before differentiating

user89138
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