- #1

NoahsArk

Gold Member

- 215

- 21

- Summary:
- Why is this a subject of it's own.

Hello. After a lot of researching, I am still not clear how the subject of differential equations is really any different from derivatives and integrals which are learned in the main part of calculus. For example:

"Population growth of rabbits:

N = the population of rabbits at any time t

r= growth rate (.01 per week in the example)

## \frac {dN}{dt} ## = the population's rate of change

Think of ## \frac {dN}{dt} ## as how much the population changes as time changes, for any moment in time.

Let us imagine the growth rate r is .01 new rabbits per week

But that is only true at a

When the population is 2,000, we get 20 new rabbits per week.

So it is better to say that the rate of change (at any instant) is the growth rate times the population at that instant:

$$ \frac {dN}{dt} = rN $$

And that is a

What am I missing? Why does the equation ## \frac {dN}{dt} ## have any strength to it, and how is this concept any different than something learned in the derivatives part of calculus? With derivatives, instantaneous rates of change, like the one in the rabbit example, are also discussed. The whole subject of differential equations seems to be reintroducing an idea taught at the beginning of calculus. I know I must be missing something fundamental given that all sources I looked at say how useful and important differential equations are.

btw- how can I preview a post before sending it? I noticed the "preview" tab now moved, but when I press it, nothing changes and I can't check if my latex is correct.

Thanks,

Joe

"Population growth of rabbits:

N = the population of rabbits at any time t

r= growth rate (.01 per week in the example)

## \frac {dN}{dt} ## = the population's rate of change

Think of ## \frac {dN}{dt} ## as how much the population changes as time changes, for any moment in time.

Let us imagine the growth rate r is .01 new rabbits per week

**for every current rabbit.**When the population is 1,000, the rate of change ## \frac {dN}{dt} ## is then 1,000 x .01 =**10 new rabbits**per week.But that is only true at a

**specific time**, and doesn't include that the population is constantly increasing. The bigger the population, the more rabbits we get!When the population is 2,000, we get 20 new rabbits per week.

So it is better to say that the rate of change (at any instant) is the growth rate times the population at that instant:

$$ \frac {dN}{dt} = rN $$

And that is a

**Differential Equation**, because it has a function,**N(t)**and its derivative." https://www.mathsisfun.com/calculus/differential-equations.htmlWhat am I missing? Why does the equation ## \frac {dN}{dt} ## have any strength to it, and how is this concept any different than something learned in the derivatives part of calculus? With derivatives, instantaneous rates of change, like the one in the rabbit example, are also discussed. The whole subject of differential equations seems to be reintroducing an idea taught at the beginning of calculus. I know I must be missing something fundamental given that all sources I looked at say how useful and important differential equations are.

btw- how can I preview a post before sending it? I noticed the "preview" tab now moved, but when I press it, nothing changes and I can't check if my latex is correct.

Thanks,

Joe