Exponential Integration Question

[GreenPrint]

Why is e^{\int \frac{dt}{t}} = e^{ln|t|} = t as apposed to |t|? I don't understand what happened to the absolute value operator. Thanks for any help.

I understand that e^{x}>0. Is this the justification? But I don't understand why you can't have a negative t in e^{ln|t|} because you would take the absolute value of a negative number.

 

[Mark44]

Why is e^{\int \frac{dt}{t}} = e^{ln|t|} = t as apposed to |t|?

It should be |t|, as you thought.

It's possible that there is some other context that you're not including, in which t is assumed to be positive. In that case, |t| = t.

I don't understand what happened to the absolute value operator. Thanks for any help.

I understand that e^{x}>0. Is this the justification? But I don't understand why you can't have a negative t in e^{ln|t|} because you would take the absolute value of a negative number.

 

[GreenPrint]

If I had the differential equation

\frac{dy}{dt} + \frac{y}{t} = 5

Then using integration factors

y = \frac{5∫e^{\int \frac{dt}{t}}dt}{e^{\int \frac{dt}{t}}} = \frac{5∫e^{ln|t|}dt}{e^{ln|t|}} = \frac{5∫|t|dt}{|t|}

I'm unsure how to proceed without ignoring the absolute value functions but it appears ignoring them seems to be just fine for whatever reason

 

[GreenPrint]

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