Solving Volterra Equations: Integrating for Solutions

[sara_87]

I am reading an example from a book on Volterra equations but there's one point i don't understand. the book says:
for certain types of linear integrodifferential equations, the reduction can be made directly by integration. consider for instance, the linear equation:

$$f'(t) - \int^{t}_{0}k(t,s)f(s) ds=g(t)$$,
with f(0)=f₀. Integrating this we get:
$$f(t)- \int^{t}_{0}$$$$\int^{T}_{0}k(T,s)f(s) dsdT=G(t)$$

I don't understand how that can be derived by integration.

 

[tiny-tim]

Hi sara_87! smile:

It's the "fundamental theorem of calculus" …

the derivative of ∫_a^t f(x) dx = f(t)

 

[sara_87]

but where did the T and G come from?

 

[tiny-tim]

If that's g'(t) in the top line, then G is simply g.

T is just a "dummy" value of t.

Try differentiating the bottom equation, and you'll see that it works!​