Discussion
The solution of the Volterra integral equation of first kind
\(f(t) = \displaystyle\int_{0}^{t} e^{t-s}\, u(s)\, ds,\)
where \(f(t)\) is known function, is :
Answer: B
The solution is \(u(t) = f'(t) - f(t)\).
Write \(f(t) = e^{t}\int_{0}^{t} e^{-s}u(s)\,ds\) and differentiate:
\[ \begin{aligned} f'(t) &= e^{t}\int_{0}^{t} e^{-s}u(s)\,ds + e^{t}e^{-t}u(t) \\ &= f(t) + u(t) \end{aligned} \]
so \(u = f' - f\). Check with Laplace transforms: \(F(s) = U(s)/(s - 1)\), so \(U = (s - 1)F = sF - F\), which is the transform of \(f' - f\) when \(f(0) = 0\) (as the equation forces). The options with a factor \(t\) do not come from this differentiation.
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