How to Solve a Diff. Eq. with Proportional Change?

[DeG]

I apologize in advance for the formatting. I don't know how to put in all the symbols and what not so I am using only what is on my keyboard.
I am trying to work out this equation and I'm stumped. Also I haven't taken a diff. eq. class yet, but plan to real soon.
I'm working with the following:
df=int(f/r^2,dr,r1,r2). I am trying to figure out the change in f between r1 and r2, but the trouble I'm having is that df is proportional to f and I just don't know how to evaluate it.
Any help would be greatly appreciated. Thanks.

 

[pmsrw3]

I'm having trouble understanding this. Tell me if this is what you want to write.

$$df=\int_{r_1}^{r_2}{\frac{f}{r^2}dr}$$

If it is, then I don't understand it. What is f? And how come the left-hand-side is a differential? And what do you mean by "df is proportional to f"?

Perhaps you're thinking that "df" means d times f? If so, that's a big source of confusion (it doesn't), but I still don't understand the equation.

 

[DeG]

Yeah, that is what I was trying to write. By df I meant the change in f. Maybe it is supposed to be df/dr, but I don't exactly know how to interpret the problem I am trying to solve. By df is proportional to f, I meant that the change in f depends on the current value of f.

 

[pmsrw3]

Yeah, that is what I was trying to write. By df I meant the change in f. Maybe it is supposed to be df/dr, but I don't exactly know how to interpret the problem I am trying to solve.

I would recommend (strongly!) against using "df" to refer to the change in f. "df" is universally used to mean the differential of f, just like dr in your integral is the differential of r. If you use df to mean $\Delta f$, you're going to confuse everyone.

What's the problem you're trying to solve? What is f? Is it a function of r?

By df is proportional to f, I meant that the change in f depends on the current value of f.

Ah. OK. That is not what "is proportional to" means. Although you'll hear people using it sloppily just to indicate a relationship, to a physicist or a mathematician "y is proportional to x" means "y equals a constant times x".

 

[DeG]

Isn't a differential simply incremental change? I suppose it is misused because I meant absolute change, but I don't see why incremental change wouldn't work. Is f a function of r? Clearly, since it's change depends on the values of r. Sorry for the confusion and any trouble.

 

[pmsrw3]

Isn't a differential simply incremental change? I suppose it is misused because I meant absolute change, but I don't see why incremental change wouldn't work.

A differential isn't a true number with a value. Strictly speaking it doesn't have any meaning outside of its syntactic use in expressions, but if you want to think of it as having a value, you should imagine it as being tiny, smaller than any number. Now, the right-hand-side of your equation is an integral with a definite value. It doesn't make any sense to equate a differential to it.

Is f a function of r? Clearly, since it's change depends on the values of r.

Well, but you didn't tell us that it does. Your equation doesn't show the change in f depending on values of r. In

$$\Delta f = \int_{r_1}^{r_2}{\frac{f}{r^2}dr}$$

r is just a dummy variable of integration. It is is perfectly consistent with f not being a function of r. And typically (this is just another matter of using the notation others use, to help communication) a function of r would be written f(r) to make the relationship clear.

 

[nickalh]

Are you making up an equation to solve? Is this an equation you're writing as part of a word problem. The quoted statement, A is a word problem. Please give us the original question verbatim from the book or other source.

Did you mean A. "The derivative of f with respect to __?__ is proportional to f"?

This would translate to
B.
df/d? = k * f.
where k is a constant.

Obviously, it should have some other variable instead of the question mark.

C.i. In the introductory, differential equation courses I have seen, integrals and derivatives on the same line of an equation are rare. It appears to be more like a common mistake I see Calculus students make. When working an integral exercise, or finding a derivative, students often aren't clear, when to drop the integral sign or when to change from f to df/dx.

C.ii. When to drop the integral sign:
The integral sign indicates an operation, a mathematical verb, like square root or ln. These are things we do: integrate g, take square root of x, take log of x. When you've found the integral, meaning you've done what the integral asks for, then we can drop the integral sign or use the vertical bar to indicate evaluate at the limits of integration. For integrals, this often requires algebraic manipulation, like substitution or rearrangement so we can do the actual integration on one line.

Please excuse me if C. is slightly off-topic.

Back to B. from above.
Once you've got the equation, we're supposed to be solving clear in your head, then we'll need an integral (possibly two for some methods, like integrating factor) somewhere when finding the solution. At that point the integral will cancel out the derivative.

df/dz = k * f
becomes
f = ∫ k*f dz
One solution to this is f= e^(k z)

D. To prepare for differential equations, first know integrals thoroughly, because every diff. eq. will involve an integral. But knowing integrals requires knowing derivatives.