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Following on from my question here about the integral equation

$y(x)=1+\int^{x}_{0}(\tanh s)y(s)ds$

we now look to appeal to Picard's Theorem.

Let $\{y_n\}_{n \geq 0}$ be the sequence of Picard Approximations for which

$y_0(x)=1$

$y_{n+1}=1+\int^{x}_{0}\tanh(s)y_n(s)ds \quad \quad (n \geq 0)$

We're asked to prove that

$y_n(x)=\sum_{k=0}^{n} \frac{1}{k!} (\log \cosh x)^k$ and that $y_n \longrightarrow y$ as $n \longrightarrow \infty$.

We know the solution $y$ to be the function $y:=\cosh(x)$.

Any help with determining how this works would be very appreciated. Regards and best, as always. MM

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    @Jon: Don't know how I didn't spot that. Thanks.2012-01-12

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I repeat the comment as an answer being it correct.

Just iterate putting $y_0(s)=1$ and using the fact that $\int ds\tanh(s)=\log\cosh s+C$.