I got the following equation as solution of the soap film surface variational problem
$u(x)= \frac{1}{c_1}\cosh(c_1x+c_2)$
$u(0)=r$, $u(b)=r$
My problem is I can't solve for $c1$ and $c2 $
$c_1r= \cosh(c_2)$
$c_1r= \cosh(c_1b+c_2)$
I am supposed to express $u$ in terms of$ b$ and then analize the limiting behavior for $b>>r$. One of the constants is supposed to be found explicitly and the other one implicitly. I already tried everything. Any help will be much appreciated
If you look at your system of equations, you can set them equal to get $$\cosh(c_2)=\cosh(c_1b+c_2)$$
To solve from here note that $\cosh(x)=\cosh(y)$ admits two solutions, either $x=y$ or $x=-y$. Can you continue from here?
Edit: So $x=y$ does not give a solution and after plugging in $c_2=-c_1b/2$ we get the implicit equation of $c_1$: $$c_1r=\cosh(-c_1b/2)=\frac{1}{2}\left(e^{c_1b/2} + e^{-c_1b/2}\right)$$ Now notice that the exponential is growing much faster than the linear term so it seems there is no solution for $b >> r$. This can be formalized by estimating say $$ 2c_1r > e^{c_1b/2} > c_1b/2$$ which gives $r > b/4$. In other words, if $ b >> r$ the equation has no solution. This corresponds to physical intuition from the soap bubbles. What do you expect the soap film to do if you increase $b/r$?