Integrating Factor: Homework Help

[EastWindBreaks]

Homework Statement

hello, I was reading through the textbook and I have a hard time to understand this part:

Homework Equations

The Attempt at a Solution

haven't been dealing with derivatives for a while, i don't understand how it got ln |u(t)| from the first equation.
Am I treating the derivative as a fraction here? how does u(t)/u(t) = ln |u(t)| ?

any help is greatly appreciated

 

[Simon Bridge]

The first equation is more usually written: $$\frac{1}{\mu}\frac{d\mu}{dt} = \frac{1}{2}$$ ... that help?
The $\ln|\mu|$ part comes from the chain rule.

I think the author is specifically trying not to treat the Leibnitz notation as a fraction, since the usual way to proceed from (1) would be to write: $$\frac{d\mu}{\mu} = \frac{dt}{2}$$ ... which is considered sloppy notation.

 

[Mark44]

The first equation is more usually written: $$\frac{1}{\mu}\frac{d\mu}{dt} = \frac{1}{2}$$ ... that help?
The $\ln|\mu|$ part comes from the chain rule.

I think the author is specifically trying not to treat the Leibnitz notation as a fraction, since the usual way to proceed from (1) would be to write: $$\frac{d\mu}{\mu} = \frac{dt}{2}$$ ... which is considered sloppy notation.

"Sloppy" is in the eye of the beholder. The differential equation in the first paragraph above is separable, as can be seen in the equation immediately above. Separation of variables is a standard technique in solving differential equations. In this technique, which is one of the first taught in a course on ODE, derivatives in Leibniz form are treated as fractions.

The next step is to integrate both sides, which yields $\ln|\mu| = \frac 1 2 t + C$.

 

[Simon Bridge]

It's the way that produces the fewer headaches.

 

[Mark44]

It's the way that produces the fewer headaches.

At the very least, that's debatable. If, for example, $y = t^2$, then we can write the derivative of y with respect to t as $\frac{dy}{dt} = 2t$ or we can write the differential of y as $dy = 2t~dt$. The latter form is used all the time in substitutions for integration problems.

Also, as I mentioned before, separation of variables is a standard technique in virtually all differential equations textbooks, and one that is usually the first technique presented.

 

[Simon Bridge]

I was unclear - I was agreeing with you.