I have trouble with solving Volterra integral equation by using differentiation method
$\varphi(x)=x-\int_0^x e^{x-t}\varphi(t)dt.$
so, I guess I need to find derivative of $\varphi(x)$, but what's next? Can you please help with algorithm. I also have problems with finding derivative of integral. Can you tell how this is done?
The answer of the task is $\varphi(x) = x - \frac{x^2}{2}$
Rewrite the desired expression as $$\phi(x)e^{-x} = xe^{-x} - \int_{0}^{x}e^{-t}\phi(t)$$ Differentiating both sides, we get $$e^{-x}((\phi(x))'-\phi(x)) = e^{-x}(1-x)-e^{-x}\phi(x)$$ $$\therefore (\phi(x))'-\phi(x) = 1-x - \phi(x)$$ $$\therefore (\phi(x))'= 1-x$$ From this, $$\phi(x) = x-\frac{x^2}{2} + C$$ where C is some constant which can be found by putting $x=0$ in the initial expression $$\therefore \phi(x) = x - \frac{x^2}{2}$$
Hope this helps.