Need to change variables in equations with cosh.

Question

i have these five functions:

$x=\tau \cosh(s)$

$q=\tau \sinh(s)$

$y= \sinh(s)$

$p= \cosh(s)$

$u= 1/2*\tau*\cosh(2s)+1/2*\tau$

I need to write $u$ in terms of $x$ and $y$

I know the answer is $u=x*\sqrt{y^2+1}$

This is part of a nonlinear PDE problem,solved with characteristics, but I dont know how to change the variables back.

Answer 1

Do dome hyperbolic trigonometry: as $\cosh2s=2\cosh^2s-1$, we can rewrite $u$ as $$u= \tau\cosh^2s-\frac12\tau+\frac12\tau=x\cosh s.$$ On the other hand, $\;\cosh^2s-\sinh^2s=1$, whence, as $\cosh s\ge 1>0$ for all $s$, $$\cosh s=\sqrt{\sinh^2s+1}=\sqrt{y^2+1}.$$

Answer 2

$$ u= \tau*(1+\cosh(2s))/2 = \tau \cosh^2 s = \tau (1+\sinh^2 s) = \tau (1+y^2) $$ $$ =\frac {x}{\cosh s}(1+y^2) = x \frac {(1+y^2) }{\sqrt{1 + \sinh^2 s}}$$ $$ = (1+y^2) \frac{x}{ \sqrt{1+y^2} } = x \sqrt{1+y^2}. $$