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Differential Equation

So I had this question which I got from the new spec sample paper. There is this differential equation which you have to solve.

Unparseable latex formula:

\[\frac{dn}{dt}=0.1n(1-\frac{n}{50})\]



It says show that he solution general solution to the differential equation can be written in the form where A is an arbitrary positive constant.

Unparseable latex formula:

\[n]



You have to find the general solution to it. I have tried multiple ways and tried to see if any work but with no luck. Need someone to help me out here
Post your working out.

Also note:
If dd is a constant then ede^{d} is also a constant.
You can seperate it into:
10n(1n50)dy=dx\int \frac{10}{n(1-\frac{n}{50})}dy=\int dx
then integrate after using partial fractions on the LHS.
Original post by Edgemaster
You can seperate it into:
10n(1n50)dy=dx\int \frac{10}{n(1-\frac{n}{50})}dy=\int dx
then integrate after using partial fractions on the LHS.


That is exactly what I was thinking. Anyway, I did the following and still can't seem to get the answer. Here is what I did

Unparseable latex formula:

\[\frac{dn}{dt}=0.1n-\frac{0.1n^{2}}{50}\]



Unparseable latex formula:

\[500\int]



Unparseable latex formula:

\[10\left [ ln (n) + ln (1-n) \right ] = t + c\]



Somehow can't get the answer because either I have done it wrong or I can't evaluate and simplify.
Original post by y.u.mad.bro?
That is exactly what I was thinking. Anyway, I did the following and still can't seem to get the answer. Here is what I did

Unparseable latex formula:

\[\frac{dn}{dt}=0.1n-\frac{0.1n^{2}}{50}\]



Unparseable latex formula:

\[500\int]



Unparseable latex formula:

\[10\left [ ln (n) + ln (1-n) \right ] = t + c\]



Somehow can't get the answer because either I have done it wrong or I can't evaluate and simplify.


It appears you either did the partial fractions incorrectly or the integrating incorrectly.
Original post by y.u.mad.bro?
That is exactly what I was thinking. Anyway, I did the following and still can't seem to get the answer. Here is what I did

Unparseable latex formula:

\[\frac{dn}{dt}=0.1n-\frac{0.1n^{2}}{50}\]



Unparseable latex formula:

\[500\int]



Unparseable latex formula:

\[10\left [ ln (n) + ln (1-n) \right ] = t + c\]



Somehow can't get the answer because either I have done it wrong or I can't evaluate and simplify.


Can you show us the complete working for partial fractions?
Original post by Edgemaster
Can you show us the complete working for partial fractions?


Unparseable latex formula:

\[\frac{1}{n(50-n)}=\frac{A}{n}+\frac{B}{50-n}\]



Unparseable latex formula:

\[n]


Unparseable latex formula:

\[n]



Unparseable latex formula:

\[\frac{1}{n(50-n)}=\frac{1}{50n}+\frac{1}{50(50-n)}\]

Original post by y.u.mad.bro?
Unparseable latex formula:

\[\frac{1}{n(50-n)}=\frac{A}{n}+\frac{B}{50-n}\]



Unparseable latex formula:

\[n]


Unparseable latex formula:

\[n]



Unparseable latex formula:

\[\frac{1}{n(50-n)}=\frac{1}{50n}+\frac{1}{50(50-n)}\]




Remember that you have 500 outside of the integral. Multiplying this through gives:

10n+1050n dn\displaystyle\int \frac{10}{n}+ \frac{10}{50-n}\ dn
(edited 5 years ago)
Original post by IWantIPods
Remember that you have 500 outside of the integral. Multiplying this through gives:

10n+1050n dn\displaystyle\int \frac{10}{n}+ \frac{10}{50-n}\ dn


I did that if you look at my first working out I posted. I just don't know what I've done wrong here or whats wrong in the integration.
Original post by y.u.mad.bro?
I did that if you look at my first working out I posted. I just don't know what I've done wrong here or whats wrong in the integration.


1050n=10×150n=10f(n)f(n)\frac{10}{50-n} = -10\times \frac{-1}{50-n} = -10\frac{f'(n)}{f(n)}


Integrating this gives:

10ln(f(n)) -10\ln(f(n))
(edited 5 years ago)
Original post by IWantIPods
1050n=10×150n=10f(n)f(n)\frac{10}{50-n} = -10\times \frac{-1}{50-n} = -10\frac{f'(n)}{f(n)}


Integrating this gives:

10ln(f(n)) -10\ln(f(n))


Ah, I see what I did wrong there. Let me redo and see what I get.
nvm. Still not there. I get somewhere like

[tex]\[t="ln(n)^{10}-ln(50-n)^{10}\" +="+" c=""]

Unparseable latex formula:

\[e^{t}+c]



Unparseable latex formula:

\[(e^{t})^{0.1}+c]



I still cant get the answer -_-' Something else I am missing?
[QUOTE="y.u.mad.bro?;77219968"]nvm. Still not there. I get somewhere like

Unparseable latex formula:

\[t="ln(n)^{10}-ln(50-n)^{10}\" +="+" c=""][br][br][tex]\[e^{t}+c][/tex][br][br][tex]\[(e^{t})^{0.1}+c][/tex][br][br]I still cant get the answer -_-' Something else I am missing?


Your transition from line 1 to line 2 is incorrect.

I think you should start with it in the form:

t+c=10ln(n)10ln(50n)t+c=10\ln(n)-10\ln(50-n)

then using the fact that:

mln(a)mln(b)=mln(ab)m\ln(a)-m\ln(b)=m\ln(\frac{a}{b})

then anti-log both sides of the equation.
Original post by IWantIPods
Your transition from line 1 to line 2 is incorrect.

I think you should start with it in the form:

t+c=10ln(n)10ln(50n)t+c=10\ln(n)-10\ln(50-n)

then using the fact that:

mln(a)mln(b)=mln(ab)m\ln(a)-m\ln(b)=m\ln(\frac{a}{b})

then anti-log both sides of the equation.


I did do that the first time around but didn't get the answer so I used the other one which was again wrong.

Unparseable latex formula:

\[ln(\frac{n}{50-n})]



Unparseable latex formula:

\[\frac{n}{50-n}=e^{\frac{t+c}{10}}\]



but I can't seem to simplify this to the required answer.
Original post by y.u.mad.bro?
I did do that the first time around but didn't get the answer so I used the other one which was again wrong.

Unparseable latex formula:

\[ln(\frac{n}{50-n})]



Unparseable latex formula:

\[\frac{n}{50-n}=e^{\frac{t+c}{10}}\]



but I can't seem to simplify this to the required answer.


Let A=ec10A=e^{\frac{c}{10}}
Original post by IWantIPods
Let A=ec10A=e^{\frac{c}{10}}


I did this but I don't think it makes a big difference.

Unparseable latex formula:

\[n]



Unparseable latex formula:

\[n(1+e^{\frac{1}{10}}e^{\frac{c}{10}})]



Therefore,
Unparseable latex formula:

\[n]

(edited 5 years ago)
Original post by y.u.mad.bro?
I did this but I don't think it makes a big difference.

Unparseable latex formula:

\[n]



Unparseable latex formula:

\[n(1+e^{\frac{1}{10}}e^{\frac{c}{10}})]



Therefore,
Unparseable latex formula:

\[n]


Look at what the question is asking you to prove. The numerator is 50A where A is a constant. Just divide the numerator and the denominator by et10e^{\frac{t}{10}}.

Also, in the denominator, you should have et10e^{\frac{t}{10}} not e110e^{\frac{1}{10}}.
Original post by IWantIPods
Look at what the question is asking you to prove. The numerator is 50A where A is a constant. Just divide the numerator and the denominator by et10e^{\frac{t}{10}}.

Also, in the denominator, you should have et10e^{\frac{t}{10}} not e110e^{\frac{1}{10}}.


My bad. I mistyped that but I see what you mean. Didn't read the question properly.

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