I'm trying to make sense of this solution but I get lost after line 13. Can anyone explain to me what happens here? I have no idea how the cosh fraction results from what came above it or what is happening as the solution progresses from there.
I also don't know where Bn came from or how the e expression integrates to the result on line 17
Edit:
Equations (1-4) are the problem to consider. Equations (5-7) are the reduction to single variable equations to solve. The $x$-equation seems to more familiar, which leaves the $y$-equation to consider.
Consider the equation $f'' - p^2 f = 0$, where $f=f(y)$ with the condition $f'(1)=0$ and $p = n \pi$. The solution, for the general case is, $$f(y) = C \, \cosh(p y) + D \, \sinh(p y).$$ Applying the condition leads to $$f'(y) = p \, (C \, \sinh(p y) + D \, \cosh(p y))$$ and $f'(1) = 0 = p \, (C \, \sinh(p) + D \, \cosh(p))$ or $$D = - C \, \frac{\sinh(n \pi)}{\cosh(n \pi)}.$$ Now, $$f(y) = C \, \left(\cosh(n \pi y) - \frac{\sinh(n \pi)}{\cosh(n \pi)} \, \sinh(n \pi y) \right)$$ and now use $\cosh(x - y) = \cosh(x) \cosh(y) - \sinh(x) \sinh(y)$ to obtain the form given in the problem (and in this solution).
Original:
Equation (10) gives the form of the solution that satisfies the pde. From (11-13) it is determined that $A = 0$, $\alpha = n \pi$ (not $\alpha$ of the boundary conditions), and $$C \, \sinh(n \pi) = - D \, \cosh(n \pi).$$ Now, The solution to the $y$-variable is $f(y) = C \, \cosh(n \pi y) + D \, \sinh(n \pi y)$ which can be seen as $$f(y) = C \, \left( \cosh(n \pi y) - \frac{\sinh(n \pi) \sinh( n \pi y)}{\cosh(n \pi)} \right) = \frac{\cosh(n \pi (1- y))}{\cosh(n \pi)}.$$ Since $A = 0$ then the solution is seen as a series of sines in the variable $x$ and takes the form $$u(x,y) = \sum_{n=1}^{\infty} \frac{B_{n}}{\cosh(n \pi)} \, \sin(n \pi x) \, \cosh(n \pi (1-y)).$$ Applying the final condition, $u(x,0) = \alpha \, e^{- \beta x}$ leads to $$\alpha \, e^{- \beta x} = \sum_{n=1}^{\infty} B_{n} \, \sin(n \pi x).$$ The coefficients $B_{n}$ is given by \begin{align} B_{n} &= \alpha \, \int_{0}^{1} e^{- \beta x} \, \sin(n \pi x) \, dx \\ &= \frac{n \pi \alpha}{\beta^2 + n^2 \pi^2} \, (1 - (-1)^n \, e^{-\beta}). \end{align}
From all of this the solution then takes the form $$u(x, y) = \sum_{n=1}^{\infty} \frac{n \pi \alpha}{\beta^2 + n^2 \pi^2} \, (1 - (-1)^n \, e^{-\beta}) \, \sin(n \pi x) \, \cosh(n \pi (1-y)).$$