# Thread: Help with deriving an equation involving hyperbolic functions...

1. ## Help with deriving an equation involving hyperbolic functions...

Hey guys, I'm in System Dynamics and have been given a fairly fundemental problem to work on.

It states: $x(t) = Acosh(pt) + Bsinh(pt)$ can also be written $x(t) = Ce^{pt} + De^{-pt}$. It then instructs me to derive equations for C and D in terms of A and B. I know this is pretty fundemental, I also know that $e^{pt} = cosh(pt) + sinh(pt)$, and $e^{-pt} = cosh(pt) - sinh(pt)$ but I do not want to confuse deriving an equation with substitution.

I'm not looking for an answer or anything like that, I just need someone or several people to put me on the right path. I'm not quite sure where to start. Thank you.

2. ## Re: Help with deriving an equation involving hyperbolic functions...

Originally Posted by DaveDammit
It states: $x(t) = Acosh(pt) + Bsinh(pt)$ can also be written $x(t) = Ce^{pt} + De^{-pt}$. It then instructs me to derive equations for C and D in terms of A and B.
If I were you, I would work directly from the definitions.
$\cosh(t)=\frac{e^t+e^{-t}}{2}~\&~\sinh(t)=\frac{e^t-e^{-t}}{2}$

3. ## Re: Help with deriving an equation involving hyperbolic functions...

Ok, thank you for the head start. So now I have just a couple of questions:

1.) Because these equations are based on the displacement of a point, should I be looking at taking a derivative?
2.) Since he's giving me the values for two variables and there's two equations, is this going to be a two equations, two unknowns deal?

4. ## Re: Help with deriving an equation involving hyperbolic functions...

Originally Posted by DaveDammit
Ok, thank you for the head start. So now I have just a couple of questions:
1.) Because these equations are based on the displacement of a point, should I be looking at taking a derivative?
2.) Since he's giving me the values for two variables and there's two equations, is this going to be a two equations, two unknowns deal?.
I have no idea what you are doing here.
I have no idea what "displacement of a point" could even mean.
Sorry, but I know nothing about applying mathematics.

5. ## Re: Help with derivation using hyperbolic functions please...

In general a function of real variable $f(x)$ can be devided into its 'even part' and 'odd part' as...

$f(x)= f_{e}(x)+f_{o}(x)$ (1)

... where...

$f_{e}(x)= \frac{f(x)+f(-x)}{2}$

$f_{o}(x)= \frac{f(x)-f(-x)}{2}$ (2)

Now if $f(x)= e^{x}$ we have...

$f_{e}(x)= cosh x$

$f_{o}(x)= sinh x$ (3)

... and if $f(x)= e^{-x}$ we have...

$f_{e}(x)= cosh x$

$f_{o}(x)= - sinh x$ (4)

Using (1),(2),(3) and (4) You are able to pass from A and B to C and D and vice versa without diffculties...

Kind regards

$\chi$ $\sigma$

6. ## Re: Help with deriving an equation involving hyperbolic functions...

Originally Posted by DaveDammit
Ok, thank you for the head start. So now I have just a couple of questions:

1.) Because these equations are based on the displacement of a point, should I be looking at taking a derivative?
2.) Since he's giving me the values for two variables and there's two equations, is this going to be a two equations, two unknowns deal?