Separable differential equation

[M55ikael]

Homework Statement

I'm trying to solve this equation:

dN/dt = k(10 000 - N(t))

The attempt at a solution

dN/10 000 - N(t) = k dt

I integrate, and I'm left with:

- ln l10 000 - N(t)l = kt +C

I raise both sides with e. I don't know what e^- ln(xxx) is so I multiply both sides with -1

l10 000 - N(t)l = e^(-kt) * e^(-C) = -Ce^(-kt)

I now get the following:
N(t) = -10 000 - Ce^(-kt)

However the correct solution according to my book is:
N(t) = (+)10 000 - Ce^(-kt)

I am left with three questions:

1. What is e^- ln(...)?
2. Was I correct in making letting l10 000 - N(t)l = 10 000 + N(t) ?
3. What did I do wrong?

Thanks!

 

[hunt_mat]

You're almost there, when you took exponentials you should have obtianed:
$$10^{4}-N(t)=Ae^{-kt}$$
I don't know what you said $$e^{-C}=-C$$ and hence:
$$N(t)=10^{4}-Ae^{-kt}$$
and you would use the initial conditions to find A.
To answer your questions though:
1)$$e^{-\ln X}=e^{-1\times\ln X}=e^{\ln X^{-1}}=X^{-1}$$
2) No, if N(t)<10^{4} then |10^{4}-N(t)|=10^{4}-N(t)
3) See above

 

[M55ikael]

You're almost there, when you took exponentials you should have obtianed:
$$10^{4}-N(t)=Ae^{-kt}$$
I don't know what you said $$e^{-C}=-C$$ and hence:
$$N(t)=10^{4}-Ae^{-kt}$$
and you would use the initial conditions to find A.
To answer your questions though:
1)$$e^{-\ln X}=e^{-1\times\ln X}=e^{\ln X^{-1}}=X^{-1}$$
2) No, if N(t)<10^{4} then |10^{4}-N(t)|=10^{4}-N(t)
3) See above

Wow, that was quick!
Could you explain what A is equal to? I raise - kt - C with e and get $$e^{-kt}*e^{-C}$$. How did you get A?

Also, it seems I have missed something when it comes to absolute values, could you explain why N(t) becomes negative and 10^4 becomes positive?

The initial conditions are that N(0)=200

 

[hunt_mat]

$$A=e^{-C}$$, you find this by setting t=0 and using N(0)=200. The abolsute value is defined as:
$$|x|=x\quad x\geqslant 0,|x|=-x\quad x<0$$

 

[M55ikael]

$$A=e^{-C}$$, you find this by setting t=0 and using N(0)=200. The abolsute value is defined as:
$$|x|=x\quad x\geqslant 0,|x|=-x\quad x<0$$

Thanks so much!
How do you determine N(t) is less than zero? Should I enter a value for t or something?

Furthermore, my book has taken a slightly different approach ending up with:

l N(t) - 10 000) l = Ce^(-kt)

They then conclude that N(t) = 10 000 - Ce^(-kt) because N(t) is equal to or less than 10 000. That would seem to contradict the previous statement if I interpreted the symbol $$x\geqslant$$ correctly.

$$|x|=x\quad x\geqslant 0$$

 

[hunt_mat]

I didn't, I stated that:
$$10^{4}-N(t)\geqslant 0\Rightarrow |10^{4}-N(t)|=10^{4}-N(t)$$
Mat

 

[M55ikael]

I didn't, I stated that:
$$10^{4}-N(t)\geqslant 0\Rightarrow |10^{4}-N(t)|=10^{4}-N(t)$$
Mat

I wasn't accusing you of anything, I'm only trying to figure out how this works.

How do I determine the value of N(t) if at all? I am still clueless.

Why is this true:

$$N(t)-10^{4}\geqslant 0\Rightarrow |-10^{4}+N(t)|=-(-10^{4}-N(t))$$

$$10^{4}-N(t)/geqslant 0\Rightarrow |10^{4}-N(t)|=10^{4}-N(t)$$

$$|x|=x\quad x\geqslant 0$$

It doesn't make sense to me. The first two statements seem to contradict each other IMHO.

 

[hunt_mat]

I know you weren't. I was clarifying.
$$N(t)-10^{4}\geqslant 0\Rightarrow |-10^{4}+N(t)|=-10^{4}+N(t)$$
Is true be definition as |x|=x if x>0.
Once you get rid of the modulus sign, you can just re-arrange.

 

[hunt_mat]

You know this from solving the diffential equation:
$$10^{4}-N(t)=Ae^{-kt}$$
Rearrange to obtain:
$$N(t)=10^{4}-Ae^{-kt}$$
We find A by using N(0)=200, hence:
$$200=10^{4}-A$$
Does this help at all?

 

[M55ikael]

Yes it does :-)

I think you meant$$N(t)-10^{4}< 0\Rightarrow |-10^{4}+N(t)|=-10^{4}+N(t)$$

 

[hunt_mat]

I don't think I did, I think that 10^4-N(t)>0.

 

[M55ikael]

My book states the following:
N(t) ≤ 10 000 => N(t) - 10 000 ≤ 0 => l N(t) - 10 000 l = -(N(t) - 10 000)

EDIT:
You are misunderstanding, please note that I have reversed the values. Like I said, this is the approach the book took, not me. They multiplied by -1 before integrating and got a different modulus.

 

[hunt_mat]

We're saying the same thing, try to convince youself of that fact.

 

[M55ikael]

Yup. But back to the N(t) value question. Do you take into account all the values of N(t) when solving the inequality and getting rid of the modulus?

I guess you solve the inequality for N(0) = 200 up to N(x)= 10 000

 

[hunt_mat]

You don't solve any inequalities. All I am saying that saying is that N(t)<10000, and so |10000-N(t)|=10000-N(t), from here on in the derivation goes as I said it did.