ATAR Notes: Forum

VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: samiira on July 28, 2010, 06:01:03 pm

Title: Challenging Spesh Q
Post by: samiira on July 28, 2010, 06:01:03 pm: any help much appreciated :'(
Title: Re: Challenging Spesh Q
Post by: m@tty on July 28, 2010, 06:51:25 pm: $\frac{dN}{N}=(a-b)dt$

$\implies \int\frac{dN}{N}=\int(a-b)dt$

$\implies \log_e|N|+C_1=(a-b)t+C_2$

$\implies N=e^{(a-b)t+C_2-C_1}$

$N=Ae^{(a-b)t}$ where $A=e^{C_2-C_1}$ ...

I have done basically none of the questions where you are asked to "solve" a differential, so I'm not sure if they want, for example:

$\frac{dN}{dt}=N$ for the first one

Or

$N=Ae^t$ ??

Not too much help, I know. But the wording is really confusing me; I hope this helps you though, in some way. =]
Title: Re: Challenging Spesh Q
Post by: pooshwaltzer on July 28, 2010, 07:59:03 pm: Extending Matty's response...

N=Ae^[(a-b)t]; A=e^(c2-c1)

When t=0, N=a

So ... a=Ae^[(a-b)0] ... A=a

N=ae^[(a-b)t]

Limitation = change in population is independent of time, ie. a constant, which may not be a realistic expectation and carries practical implications.
Title: Re: Challenging Spesh Q
Post by: samiira on July 31, 2010, 11:36:47 am: Thanks for that.. wud u know how to do this one
Title: Re: Challenging Spesh Q
Post by: m@tty on July 31, 2010, 04:20:11 pm: $\frac{dN}{dt}=(k-cN)N$

$\int dt=\int\frac{dN}{(k-cN)N}$

And, splitting into partial fractions:
Let $\frac{1}{(k-cN)N}=\frac{A}{N}+\frac{B}{k-cN}$

$\implies 1=A(k-cN)+BN$

$\therefore A=\frac{1}{k}$ ; $B=\frac{c}{k}$

$\therefore \frac{1}{(k-cN)N}=\frac{1}{kN}+\frac{c}{k(k-cN)}$

$\therefore t+C_1=\int\left[ \frac{1}{kN}+\frac{c}{k(k-cN)}\right]dN$

$t+C_1=\frac{1}{k}\log_e|N|-\frac{1}{k}\log_e|k-cN|+C_2$

I've got to go, but that seems correct so far.
Title: Re: Challenging Spesh Q
Post by: Martoman on July 31, 2010, 06:11:15 pm: http://img812.imageshack.us/f/helpw.jpg/

That should walk you through all you need.
Title: Re: Challenging Spesh Q
Post by: samiira on July 31, 2010, 11:05:29 pm: THANK YOUUUU!!!!!.. :) :) :) :)