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Find a formula for y[x] if it is known that the instantaneous growth rate y'[x] is proportional to y[x] and it is known that y[0]=5 and y[5.7]=10.
Does the value of y[x] double every 5.7 years?

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The section they're in is Differential Equations and the most common one they use is y'[x] = r y[x], where y[x]=k E^rx (k and r being constants). I'm having trouble explaining this problem to my student. Any (constructive) help is welcome.

The wording of the problem means that we have an equation similar to:

dy/dx=r*y

For simplicity (and to make the problem yield a meaningful solution), we want to make sure that anything that directly depends on y is integrated with respect to y, and anything that directly depends on x is integrated with respect to x. We accomplish this by a series of algebraic manipulations. In this case, we first divide both sides by y, yielding:

dy/dx/y=r

Then, we multiply both sides by dx, so that the other side will be the integral w.r.t. x, yielding:

dy/y=r*dx.

Now that we've done that, we simply integrate both sides.

The student should remember from earlier that Integral[du/u]=ln|u|.

So...ln|y|=Integral[r*dx]

Since r has no dependence on x, we can remove it from the integral, yielding:

ln|y|=r*Integral[dx]

Which trivially yields:
ln|y|=r*x + C

(Remember, we need to put the constant of integration somewhere, and I put it the place that it's usually put)

Raising e to both sides, we find
e^^(ln|y|)=e^^(r*x + C)

The definition of ln (inverse of e^), yields:
|y|=e^^(r*x + C).

Since e^^(r*x + C) is always positive, the absolute value doesn't matter (you can just drop the whole thing altogether if the professor isn't being strict about that kind of thing). Using that and one of the properties of exponents, we find:
y=e^^(r*x) * e^^(C)

However, since C is just an arbitrary constant of integration, we can define a new C (might call it C prime if you're having trouble conveying this concept) which is e^^C, the substitution of which yields:
y=C*e^^(r*x)

Now, we've got two unknowns: C and r. The only way to find these is to use two known points. If you didn't have the value at x=0, you'd have to use a system of equations, but since you do, you can simply plug that in first. In this case r*0 is 0, e^^0 is 1, and 1*C=C. So C=5.

Now, the equation is of the form:
y=5*e^^(r*x)

Now, plugging in the second set of known values, we find:
10=5*e^^(r*5.7)

A bit of algebra, or a calculator, depending on the situation should see the student through solving for x.

Now, if the student compares the value of y[x+5.7] to y[x], he or she should see that using the same property of exponents we used to define our new constant, he or she can find a geometrical relationship between the two values. The common ratio will either be greater than or equal to 2, in which case the answer to the problem's question is yes, or it will be less than 2, in which case the answer is no.

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Karnaj had this to say about pies:
I'm SO glad I stopped after Calc 2.

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You're missing out on the fun stuff.

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Espio Idsavant had this to say about the Spice Girls:
I went all the way to Multivariable Calc III, and my brain still hasnt recovered after 2 years.

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I miss multivariable calc.

Calc 1 (refresher/grade improvement)
Calc 3 (grade imp)
Linear Algebra (again)
Diff eq (New one)

this is for transfer into Astronomy... which has a forced minor in Math. so feh... I'll be a math nerd after I finish...

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When the babel fish was in place, it was apparent Alek Saege said:
Wow, that's just amazing. Thank you.

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You're welcome, I'm just sharing the love

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Aw, geez, I have Alek Saege all over myself!
You're missing out on the fun stuff.

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Sorry, all that integration fried my math center permanently. I'm lucky I'm not retarded. Well, more retarded.

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Led had this to say about Duck Tales:
And my school taught basic algrebra in 12th grade I believe I am screwed for life!

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goddamn...

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Alek Saege thought about the meaning of life:
math lovers

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OMG, gross!

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Led obviously shouldn't have said:
And my school taught basic algrebra in 12th grade I believe I am screwed for life!

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Wait, you didn't have options? You were forced to one math class?