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DE book does not justify this claim

  • Thread starter 1MileCrash
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  • #1
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Homework Statement



My DE book claims that

[itex]\frac{d\mu(t)/dt}{\mu(t)}[/itex]

is equivalent to

[itex]\frac{d}{dt}ln|\mu(t)|[/itex]

But says nothing to justify this claim.

Can you?
 

Answers and Replies

  • #2
gabbagabbahey
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But says nothing to justify this claim.
Why would it? Most Authors would assume that if a student is studying differential equations, they have already mastered the basics of single variable calculus, such as computing derivatives and anti-derivatives.

This is just an application of the chain rule, combined with the fact that [itex]\frac{d}{dx} \ln|x| = \frac{1}{x}[/itex].
 
  • #3
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I see how from the bottom, the top follows by direct application of the chain rule.

But given the top, I don't know how to come to the bottom expression.
 
  • #4
Bacle2
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But that seems just like the chain rule , (though I don't know why they dropped the

absolute value): d/dt f(g(t))= f'(g(t))g'(t) , where f(t)=ln|t| g(t)=u(t). Or

maybe you can do a triple chain rule f(g(h)) , with g(t)=|t|, h(t)=u(t), and

f(t)=ln(t).
 
  • #5
gabbagabbahey
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I see how from the bottom, the top follows by direct application of the chain rule.

But given the top, I don't know how to come to the bottom expression.
Just compute the anti-derivative:

[tex]\int \frac{ \frac{d\mu}{dt} }{ \mu(t) }dt = \int \frac{d\mu}{\mu} = \ln|\mu| + C[/tex]

More importantly though, given how common a function the natural logarithm is, you need to be able to recognize its derivative when you see it. A big part of problem solving with differential equations requires you to have experience with calculating common derivatives (which is why most calculus textbooks have all those practice problems in them :wink:) and applying that experience, by recognizing derivatives of common functions when you see them.
 
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  • #6
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Remember that:
[itex]\large \frac{δ}{δt}ln|x|=\frac{1}{x}[/itex]

but since you have a another function inside the log, you need to apply the chain rule, so:
[itex]\large \frac{δ}{δt}ln|\mu(t)|=\frac{1}{\mu(t)}\cdot\frac{δ\mu(t)}{δt}[/itex]
 
  • #7
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That's not really my issue. I just like to see it written out even if it is recognizable - I can integrate and differentiate until the cows come home.

Nonetheless:

[itex]\frac{d\mu(t)/dt}{\mu(t)} = q[/itex]

Integrating both sides

[itex]ln|\mu(t)| = qt[/itex]

Then taking the derivative of both sides

[itex]\frac{dln|\mu(t)|}{dt} = q[/itex]

Therefore:

[itex]\frac{d\mu(t)/dt}{\mu(t)} = \frac{dln|\mu(t)|}{dt} = q[/itex]

Is satisfactory.
 
  • #8
vela
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That's not really my issue. I just like to see it written out even if it is recognizable - I can integrate and differentiate until the cows come home.

Nonetheless:

[itex]\frac{d\mu(t)/dt}{\mu(t)} = q[/itex]

Integrating both sides

[itex]ln|\mu(t)| = qt[/itex]
This all seems kind of pointless. Integrating ##\frac{d\mu(t)/dt}{\mu(t)}## to get ##\ln|\mu(t)|##, you've used the fact you're trying to show, so why bother? Also, the righthand side is only equal to qt if q is a constant, which it most likely isn't.
 
  • #9
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q is constant, I just put it there so I could visualize it thoroughly.
 
  • #10
gabbagabbahey
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q is constant, I just put it there so I could visualize it thoroughly.
But you defined [itex]q[/itex] as being equal to [itex]\frac{ \frac{d\mu(t)}{dt} }{\mu(t)}[/itex], so it is only constant if [itex]\frac{ \frac{d\mu(t)}{dt} }{\mu(t)}[/itex] is constant, which it isn't, in general.
 
  • #11
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It is constant, here.

If it weren't, instead of qt it would be an arbitrary integral notation of q. It doesn't matter.
 

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  • #12
gabbagabbahey
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though I don't know why they dropped the absolute value
Try computing [itex]\frac{d}{dx}\ln(x)[/itex] when [itex]x<0[/itex]. There's a reason that calculus books give the antiderivative of [itex]\frac{1}{x}[/itex] as [itex]\ln|x|+C[/itex] instead of [itex]\ln(x)+C[/itex].
 

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