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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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Just iterate putting $y_0(s)=1$ and using the fact that $\int ds\tanh(s)=\log\cosh s+C$. –  Jon Jan 12 '12 at 21:37
    
@Jon: Don't know how I didn't spot that. Thanks. –  Mathmo Jan 12 '12 at 21:39

1 Answer 1

up vote 1 down vote accepted

I repeat the comment as an answer being it correct.

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

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