I am currently studying a course on mathematical methods for physicists. I am trying to solve the following integral equation:
$$y(x)=x+\int_0^1(1+xt)y(t)dt $$
with the Neumann series method. This exercise is taken from Arfken and Weber "Mathematical Methods for Physicists", seventh edition, chapter 21, exercise 21.3.2. I have already found the solution, which is $y(x) = -2$, with the separable-kernel tecnique. However, when trying to find a solution of the type:
$$ y(x) = \sum_{n=0}^{\infty}\lambda^ny_n(x)$$
I keep getting linear polynomials with "random" coefficients (in the sense that I don't find a pattern or recursive relation). Since the solution is a constant, I think this series should truncate in the first term. But $\lambda=1$, so all the orders are in fact the same. I don't really get what I should be doing. Any ideas? Thank you in advance.
Well, your linear polynomials are by no means "random". The integral operator takes a polynomial $a+b\,x$ to another polynomial $c+d\,x$ where $\begin{bmatrix}c \\d\end{bmatrix}=\begin{bmatrix}1 & \frac12\\\frac12 & \frac13\end{bmatrix}\begin{bmatrix}a \\b\end{bmatrix}$. Powers of that operator correspond to powers of the matrix. The eigenvalues of $\begin{bmatrix}1 & \frac12\\\frac12 & \frac13\end{bmatrix}$ are $\displaystyle\frac{4\pm\sqrt{13}}6$, and the powers of the matrix may grow like powers of the biggest eigenvalue, so the Neumann series can't converge for $\displaystyle\lambda\ge\frac6{4+\sqrt{13}}\approx0.78889744907202141376155746505900810750$.