Transform formula into another form

[fonseh]

Homework Statement

I need help when changing the formula of the yellow circled part to red-circled part ...

Homework Equations

The Attempt at a Solution

I tried , but i didnt get what the author got ... Here's my working ... Which part of my working is wrong ?

 

[Mark44]

Moved to Calculus and Beyond section.
@fonseh, many helpers won't respond if the problem statement and work are shown only in attached images.

 

[haruspex]

Homework Statement

I need help when changing the formula of the yellow circled part to red-circled part ...

Homework Equations

The Attempt at a Solution

I tried , but i didnt get what the author got ... Here's my working ... Which part of my working is wrong ?

I cannot make sense of the text either. It looks quite wrong. Certainly a du has been omitted from the right hand side, and I cannot see how that 1-u term can appear. Perhaps the N exponent is in the wrong place. Maybe some approximation has been made, but that should be stated.
Your own working has y on the right instead of dy (=y₀du).

 

[haruspex]

The closest I can get is $\frac{y_0du}{S_0}[\left(1-\frac {1}{1-u^N}\right)$ $(1-(\frac{y_c}{y_0})^Mu^{-M})^{-1}]$.
If we use the binomial approximation for the last (...)^-1 term, we get:
$\frac{y_0du}{S_0}[\left(1-\frac {1}{1-u^N}\right)$ $(1+(\frac{y_c}{y_0})^Mu^{-M})]$.
Multiplying the brackets out produces something quite close to the text. The remaining differences look like typos.

 

[fonseh]

The closest I can get is $\frac{y_0du}{S_0}[\left(1-\frac {1}{1-u^N}\right)$ $(1-(\frac{y_c}{y_0})^Mu^{-M})^{-1}]$.
If we use the binomial approximation for the last (...)^-1 term, we get:
$\frac{y_0du}{S_0}[\left(1-\frac {1}{1-u^N}\right)$ $(1+(\frac{y_c}{y_0})^Mu^{-M})]$.
Multiplying the brackets out produces something quite close to the text. The remaining differences look like typos.

Can you show your working pls ? I tried it many times , but didnt get the red part , i gt (1-(1/u)^N ) instead

 

[fonseh]

This is actually hydraulic formula , should i move it to another thread ? Or this is just purely mathematics derivation ? I'm not sure

 

[Mark44]

This is actually hydraulic formula , should i move it to another thread ? Or this is just purely mathematics derivation ? I'm not sure

It's fine here. The gist of the question is mathematical in nature.