DE book does not justify this claim

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In summary, the homework statement says that the result of the differential equation d\mu(t)/dt=r is equivalent to \frac{d}{dt}ln|\mu(t)|. However, the justification for this claim is not given.
  • #1
1MileCrash
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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?
 
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  • #2
1MileCrash said:
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
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


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
1MileCrash said:
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


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
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
1MileCrash said:
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
q is constant, I just put it there so I could visualize it thoroughly.
 
  • #10
1MileCrash said:
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
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


Bacle2 said:
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].
 

What does "DE book does not justify this claim" mean?

"DE book does not justify this claim" means that the evidence presented in a certain book is not sufficient or convincing enough to support a specific claim or argument.

How do I determine if a claim in a book is justified or not?

To determine if a claim in a book is justified or not, you should critically evaluate the evidence provided and see if it is supported by reliable sources, data, and logical reasoning.

What should I do if I come across a book that does not justify its claims?

If you come across a book that does not justify its claims, you can seek out other sources to verify the information or challenge the author's argument by presenting counter-evidence or logical arguments.

Can a book be considered valid if it does not justify its claims?

No, a book cannot be considered valid if it does not justify its claims. A valid book should present evidence and arguments that are supported by reliable sources and logical reasoning.

How can I avoid falling for unjustified claims in books?

To avoid falling for unjustified claims in books, it is important to critically evaluate the evidence presented, conduct your own research, and seek out multiple perspectives on the topic. It is also helpful to consult with experts or fact-checking websites.

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