Why e popped-up in Diff. Equation

[Suwandy]

I've read that Euler number (e) or 2.718 is a base for growth/decay rate of something.

what i don't get is, why differential equation has this number?
Suppose that we're talking about force F = m.a which has acceleration as derivative of velocity.
Why does the solution to the problem regarding that topic contains e ??

I thought e tells me there are some sort of change (positive/negative) in a system i.e. positive/negative acceleration.
Can someone show the correlation?

 

[Deleted member 4993]

I've read that Euler number (e) or 2.718 is a base for growth/decay rate of something.

what i don't get is, why differential equation has this number?
Suppose that we're talking about force F = m.a which has acceleration as derivative of velocity.
Why does the solution to the problem regarding that topic contains e ??

I thought e tells me there are some sort of change (positive/negative) in a system i.e. positive/negative acceleration.
Can someone show the correlation?

Do you know the following:

y = C₁*e^x → x = ln(y) + C where C = -ln(C₁) = ln(1/C₁)

and

\(\displaystyle \frac{d}{dx}[ln(x)] = \frac{1}{x}\)

\(\displaystyle \frac{d}{dx}[e^x] \ = \ e^x\)

 

[Suwandy]

Thank you for your reply, Subhotosh Khan.

[FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main][[/FONT][FONT=MathJax_Math-italic]e[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]][/FONT][FONT=MathJax_Main] [/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main] [/FONT][FONT=MathJax_Math-italic]e[/FONT][FONT=MathJax_Math-italic]x[/FONT]

i know only that part because it's logical. The rate of change of e is equal to scalar*e based on the power of e.

can you show me real-world example?
i learn mostly by visualizing things, so math equations hardly tells me anything.

 

[alain]

I've read that Euler number (e) or 2.718 is a base for growth/decay rate of something.

what i don't get is, why differential equation has this number?
Suppose that we're talking about force F = m.a which has acceleration as derivative of velocity.
Why does the solution to the problem regarding that topic contains e ??

I thought e tells me there are some sort of change (positive/negative) in a system i.e. positive/negative acceleration.
Can someone show the correlation?

You will see e popping out, among other numerous situations, in LDE of the first degree of the form dy/dt=c*y, the solution being y=e^(ct)+C

 

[alain]

Thank you for your reply, Subhotosh Khan.


i know only that part because it's logical. The rate of change of e is equal to scalar*e based on the power of e.

can you show me real-world example?
i learn mostly by visualizing things, so math equations hardly tells me anything.

It will appear, for example, in population growth equations, where the rate of change in a population is proportional to the population itself. If P is the population at any time, then dP/dt=kP, with solution P=e^kt + c

 

[alain]

It will appear, for example, in population growth equations, where the rate of change in a population is proportional to the population itself. If P is the population at any time, then dP/dt=kP, which solution is P=e^kt + c

 

[Suwandy]

[COLOR=#3E3E3E]It will appear, for example, in population growth equations, where the rate of change in a population is proportional to the population itself. If P is the population at any time, then dP/dt=kP, which solution is P=e^kt + c[/COLOR]

Thank you Alain for your reply. I have question, when the rate of change in population not equal to the population, will e pop-out?

 

[stapel]

It will appear, for example, in population growth equations, where the rate of change in a population is proportional to the population itself.

[W]hen the rate of change in population not equal to the population, will e pop-out?

If the rate is equal, will it then be proportional?

 

[alain]

[COLOR=#3E3E3E]It will appear, for example, in population growth equations, where the rate of change in a population is proportional to the population itself. If P is the population at any time, then dP/dt=kP, which solution is P=e^kt + c[/COLOR]

Thank you Alain for your reply. I have question, when the rate of change in population not equal to the population, will e pop-out?

The rate of change is proportionnal to c*P. So if c<>1, the rate of change is not equal but still proportionnal to the population, so yes e will pop up.