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frank0520
 one year ago
Use separation of variables to solve the differential equation: K dN/dt = r(NK)(NA) , K,r,A are constants.
frank0520
 one year ago
Use separation of variables to solve the differential equation: K dN/dt = r(NK)(NA) , K,r,A are constants.

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freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[K \frac{dN}{dt}=r(NK) (NA) \\ K dN=r(NK)(NA) dt \text{ divide both sides by } (NK)(NA) \\ \frac{K}{(NK)(NA)} dN=r dt \text{ now integrate both sides }\]

frank0520
 one year ago
Best ResponseYou've already chosen the best response.0that's the part I'm stuck on. the left hand side because the right is easy

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[\frac{K}{(NK)(NA)}=\frac{a}{NK}+\frac{b}{NA} \\ \frac{K}{(NK)(NA)}=\frac{a(NA)+b(NK)}{(NK)(NA)} \\ \frac{K}{(NK)(NA)}=\frac{(a+b)N+(aAbK)}{(NK)(NA)} \\ \text{ so we have teh system } \\ a+b=0 \\ aAbK=K \\ \text{ so the first equation gives } a=b \\ \text{ and using the first \in the second we have } \\ bAbK=K \\ \text{ solving for } b \\ b=\frac{K}{AK} \\ \text{ and so } a=b=\frac{K}{AK}\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.1what I'm trying to suggest from the above is to use partial fractions

freckles
 one year ago
Best ResponseYou've already chosen the best response.1are you there? still having trouble?

freckles
 one year ago
Best ResponseYou've already chosen the best response.1the left hand side you should be evaluating.. \[\frac{K}{AK} \int\limits \frac{1}{NK} dN+ \frac{K}{AK} \int\limits \frac{1}{NA} dN\]

frank0520
 one year ago
Best ResponseYou've already chosen the best response.0so would it be: \[\frac{ K }{ AK }\lnNK + \frac{ K }{ AK }\lnNA = r t + C\]

frank0520
 one year ago
Best ResponseYou've already chosen the best response.0Ok, Thank for the Help!

freckles
 one year ago
Best ResponseYou've already chosen the best response.1did you understand what I did above for the partial fraction part?

frank0520
 one year ago
Best ResponseYou've already chosen the best response.0Yes, when I did myself I put the r with the partial fractions, which made it harder like: \[\frac{ K }{ r(NK)(NA) } dN\] but now it more clearer with your way.
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