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: $\displaystyle x(t) = Acosh(pt) + Bsinh(pt)$ can also be written $\displaystyle x(t) = Ce^{pt} + De^{-pt}$. 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 $\displaystyle e^{pt} = cosh(pt) + sinh(pt)$, and $\displaystyle e^{-pt} = cosh(pt) - sinh(pt)$ 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.

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

Quote:

Originally Posted by

**DaveDammit** It states: $\displaystyle x(t) = Acosh(pt) + Bsinh(pt)$ can also be written $\displaystyle x(t) = Ce^{pt} + De^{-pt}$. 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.

$\displaystyle \cosh(t)=\frac{e^t+e^{-t}}{2}~\&~\sinh(t)=\frac{e^t-e^{-t}}{2}$

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?

Thanks for your help.

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

Quote:

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.

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

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

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

... where...

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

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

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

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

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

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

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

$\displaystyle f_{o}(x)= - sinh x$ (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

$\displaystyle \chi$ $\displaystyle \sigma$

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

Quote:

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?

Thanks for your help.

Based on what you have asked above you need to go and see your lecturer or tutor and check that you have the required background in maths to do this course and/or see what revision or catch-up resources are available to bring you up to speed on the maths background you need.

CB