## Help with an elimination and integration...

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masher
Posts: 821
Joined: Tue Oct 23, 2007 11:07 pm UTC
Location: Melbourne, Australia

### Help with an elimination and integration...

I'm having a bit of trouble following some maths in a paper I'm reading:

There are two equations defined as:

$\frac{d\eta}{dt} = - \psi^n$
$\frac{d\eta}{dt} = \frac{1-\psi}{\sigma^2} \frac{1}{\frac{dp(\eta)}{d\eta}}$

The paper then says:
As usual, we may now proceed to eliminate [imath]\psi[/imath] by combining the two equations and thus obtain expressions for the rate which can then integrated to yield a relationship between [imath]\eta[/imath] and [imath]t[/imath].

The paper gives an example for n=1 being:
$t=(1-\eta) + \sigma^2 p(\eta)$

For the life of me, I can't figure out how to get past the first stage of combining the equations to get rid of [imath]\psi[/imath]...

Any ideas/help?

jestingrabbit
Factoids are just Datas that haven't grown up yet
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Location: Sydney

### Re: Help with an elimination and integration...

Using just the first equation, can you write down an expression for psi that you can sub into the second?

[nb: this looks hideous.]
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Proginoskes
Posts: 313
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### Re: Help with an elimination and integration...

masher wrote:I'm having a bit of trouble following some maths in a paper I'm reading:

There are two equations defined as:

$\frac{d\eta}{dt} = - \psi^n$
$\frac{d\eta}{dt} = \frac{1-\psi}{\sigma^2} \frac{1}{\frac{dp(\eta)}{d\eta}}$

The paper then says:
As usual, we may now proceed to eliminate [imath]\psi[/imath] by combining the two equations and thus obtain expressions for the rate which can then integrated to yield a relationship between [imath]\eta[/imath] and [imath]t[/imath].

I'd set the two equations equal to each other, multiply both sides by [imath]-\psi^n \sigma^2 \,dp(\eta)[/imath] to get:
$\sigma^2 \,dp = \psi^{n}(\psi-1)\,d\eta.$
Then
$\sigma^2 p = \int \sigma^2\, dp = \int \psi^n(\psi-1)\,d\eta = \psi^n(\psi-1)\eta + C$
for some constant C. This assumes, of course, that n is a constant w.r.t. [imath]\eta[/imath] and [imath]\sigma[/imath] is a contant w.r.t. p. Otherwise, there are more equations to post.

The paper gives an example for n=1 being:
$t=(1-\eta) + \sigma^2 p(\eta)$

For the life of me, I can't figure out how to get past the first stage of combining the equations to get rid of [imath]\psi[/imath]...

Any ideas/help?

masher
Posts: 821
Joined: Tue Oct 23, 2007 11:07 pm UTC
Location: Melbourne, Australia

### Re: Help with an elimination and integration...

n and sigma are constants.

The best I can do is the following:

Equate the expressions:
$- \psi^n = \frac{1-\psi}{\sigma^2} \frac{d\eta}{dp}$
$\sigma^2 dp = - \frac{1-\psi}{\psi^n} d\eta$

We know that [imath]d\eta = - \psi^n dt[/imath].
$\sigma^2 dp = (1-\psi) dt$

Integrate both sides
$\sigma^2 p = (1-\psi)t + C$

We also know that [imath]\eta = - \psi^n t + D[/imath], and if we take n=1 (as in the given example), we find that [imath]\psi = \frac{\eta-D}{t}[/imath].

Substitute in, and get
$\sigma^2p = t + \eta - D + C$

and finally
$t = -C - \eta + \sigma^2 p$

which is a little different to the given example (unless C = -1)...

masher
Posts: 821
Joined: Tue Oct 23, 2007 11:07 pm UTC
Location: Melbourne, Australia

### Re: Help with an elimination and integration...

I can also do this:

Given that [imath]\frac{d\eta}{dt} = - \psi^n[/imath], we can see that [imath]\eta=- \psi^n t + C[/imath], and consequently, if n=1, [imath]\psi = \frac{C-\eta}{t}[/imath].

From this, it follows that
$\frac{d\eta}{dt} = \frac{1-\frac{C-\eta}{t}}{\sigma^2} \frac{d\eta}{dp}$

Canceling [imath]d\eta[/imath] and inverting both sides of the equation, we get
$dt = \frac{\sigma^2}{1-\frac{C-\eta}{t}} dp$
$\left( 1-\frac{C-\eta}{t} \right) dt = \sigma^2 dp$

Integrating both sides, we find:
$t - (C-\eta)\ln(t) = \sigma^2 p + D$

which is different to both my previous result, and the result given in the paper...

masher
Posts: 821
Joined: Tue Oct 23, 2007 11:07 pm UTC
Location: Melbourne, Australia

### Re: Help with an elimination and integration...

Got it!

Taking my first solution:
$t = -C -\eta + \sigma^2 p$

I can then use some boundary conditions I found implied in one of the figures: at t = 0, [imath]\eta =1[/imath] and p = 0. It follows from this that C does, in fact, equal -1.

Finally, in the case of n = 1,
$t=1-\eta + \sigma^2 p$.