Discussion
The Neumann series solution of the Volterra integral equation
\[ x(t) = 1 + t + \int_{0}^{t} (t - s)\, x(s)\, ds \]
is :
Answer: D
The solution is x(t) = et.
Differentiate twice:
\[ \begin{aligned} x'(t) &= 1 + \int_{0}^{t} x(s)\, ds \\ x''(t) &= x(t) \end{aligned} \]
with \(x(0) = 1\) and \(x'(0) = 1\), whose solution is \(e^{t}\). The Neumann series gives the same: starting from \(x_{0} = 1 + t\), each iteration adds the next terms \(t^{2}/2!,\ t^{3}/3!, \ldots\), building up \(\sum t^{n}/n! = e^{t}\).
\(\cos t\) and \(\sin t\) solve \(x'' = -x\), not \(x'' = x\), and \(\log t\) is not even defined at t = 0.
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