Simplification
- Thread starter Karim
- Start date
D
Deleted member 4993
Guest
Alternatively, I would first substitute
x = e^(2t + 100*c)
Solve for N - following the advice in response #2
Then de-substitute 'x'.
That will save you some ink (or electrons)
I presume this is what you got AFTER 'exponentiating'.
[imath]\frac{N}{100- N}= e^{t+100c}[/imath] .......................................incorrect equation
Multiply both sides by 100- N
[imath]N= (100- N)e^{t+ 100c}[/imath]
Distribute on the right.
[imath]N= 100e^{t+ 100c}- Ne^{t+ 100c}[/imath]
Add [imath]Ne^{t+ 100c}[/imath] to both sides
[imath]N+ Ne^{t+ 100c}= 100e^{t+ 100c}[/imath]
Factor out N on the left
[imath]N(1+ e^{t+100c})= 100e^{t+ 100c}[/imath]
Finally, divide both sides by [imath]1+ e^{t+ 100c}[/imath]
[imath]N= \frac{100e^{t+ 100c}}{1+ e^{t+ 100c}}[/imath]
[imath]\frac{N}{100- N}= e^{t+100c}[/imath] .......................................incorrect equation
Multiply both sides by 100- N
[imath]N= (100- N)e^{t+ 100c}[/imath]
Distribute on the right.
[imath]N= 100e^{t+ 100c}- Ne^{t+ 100c}[/imath]
Add [imath]Ne^{t+ 100c}[/imath] to both sides
[imath]N+ Ne^{t+ 100c}= 100e^{t+ 100c}[/imath]
Factor out N on the left
[imath]N(1+ e^{t+100c})= 100e^{t+ 100c}[/imath]
Finally, divide both sides by [imath]1+ e^{t+ 100c}[/imath]
[imath]N= \frac{100e^{t+ 100c}}{1+ e^{t+ 100c}}[/imath]
Last edited by a moderator:
I see that the moderator has marked the first equation as "incorrect equation". I presume that is because I lost the "2". It should be
[math]\frac{N}{100- N}= e^{2t+ 100c}[/math].
To fix that, add a "2" to "t" throughout so that the final result is [math]N= \frac{100e^{2t+ 100c}}{1+ e^{2t+ 100c}}[/math]
[math]\frac{N}{100- N}= e^{2t+ 100c}[/math].
To fix that, add a "2" to "t" throughout so that the final result is [math]N= \frac{100e^{2t+ 100c}}{1+ e^{2t+ 100c}}[/math]