Nasty differnation variables things

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In summary, the population P at time t grows at a rate proportional to P, represented by the differential equation dP/dt = kP. By solving this equation, the general solution can be written as P = Ak^{t}, where A and k are positive constants. This can be achieved through separation of variables and integration. The left side of the equation can be solved using the natural log rule for exponential functions.
  • #1
thomas49th
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A population grows in such a way that the rate of change of the population P at time t in days is proportional to P.

a) Write down a dfferential equation relating P and t

b) Show, by solving this equation, that the general solution of this equation may be written as [tex]P = Ak^{t}[/tex], where A and k are positive constants.a) is easy:

dP/dt = kP

b) I don't know where to start

Can someone walk me through B please :)
 
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  • #2
[tex]\frac{dP}{dt} = kP[/tex]

[tex]\Rightarrow \frac{1}{P}\frac{dP}{dt}=k[/tex]


integrate both sides w.r.t. t
 
  • #3
Can you show me how. I don't know how to intergrate this equation

Thanks :)
 
  • #4
hang on this is one of those stupid natural log ones

i know if y = a ^ x then dy/dx = a^x ln a

but how does that help?
 
  • #5
this is separation of variables?
 
  • #6
thomas49th said:
this is separation of variables?

Yes

[tex]\int \frac{1}{P}dP= \int k dt[/tex]



are you able to do the left side?
 

1. What are "nasty differnation variables things"?

"Nasty differnation variables things" is not a specific scientific term, but it could refer to variables that are difficult to control or measure in an experiment. These variables may have a significant impact on the outcome of the experiment, making it challenging to draw accurate conclusions.

2. How do you identify and control for "nasty differnation variables things" in an experiment?

Identifying and controlling for "nasty differnation variables things" requires careful planning and design of the experiment. Scientists must carefully consider all possible variables that could affect the outcome and devise methods to control or eliminate their influence. This may include using control groups, randomization, and blinding techniques.

3. What are some examples of "nasty differnation variables things"?

Examples of "nasty differnation variables things" could include factors such as human error, environmental conditions, or equipment malfunctions. These variables are often difficult to measure or control, and their impact on the experiment may be unknown or unpredictable.

4. How do "nasty differnation variables things" affect the validity of an experiment?

"Nasty differnation variables things" can significantly impact the validity of an experiment by introducing bias or confounding factors. If these variables are not properly controlled for, they can lead to inaccurate or misleading results. This can ultimately undermine the credibility and reliability of the experiment.

5. What steps can be taken to minimize the effects of "nasty differnation variables things" in an experiment?

To minimize the effects of "nasty differnation variables things," scientists can take several steps, such as conducting pilot studies, using rigorous experimental design, and carefully documenting and analyzing data. Additionally, peer review and replication of the experiment can help identify and address any potential issues with these variables.

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