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non-dimensionalisation

mt91

New member
Jul 30, 2020
15
1596460878703.png

I've got two differential equations I need to non-dimensionalise

I've managed to do the \[ dc/dT=α- μc \] with the following working out:

By letting:
\[ N=N0n \]
\[ C=C0c \]
\[ t=t0T \]

1596461095620.png

However I'm struggling with the first equation.

1596461128387.png

I'm up to here, it's just the left hand side of the equation I'm struggling to work out. Any help would be great, cheers
 

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
731
Looking at $\frac{dN}{dt}= \frac{rN}{h+ rN}- bNC$ and comparing that to the desired $\frac{dr}{dt}= \frac{n}{n+1}- \beta nc$ my first thought would be to divide the numerator and denominator of $\frac{rN}{h+ rN}$ by h. That gives $\frac{\frac{rN}{h}}{1+ \frac{rN}{h}}$. So we should let $n= \frac{rN}{h}$. With that substitution, the first equation becomes $\frac{h}{r}\frac{dn}{dt}= \frac{n}{1+ n}- \left(\frac{br}{h}\right)nC$.