# Integrating factor

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1. Feb 23, 2016

### EastWindBreaks

1. The problem statement, all variables and given/known data
hello, I was reading through the text book and I have a hard time to understand this part:

2. Relevant equations

3. The attempt at a solution
haven't been dealing with derivatives for a while, i dont 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

2. Feb 23, 2016

### 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.

3. Feb 23, 2016

### Staff: Mentor

"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 $\ln|\mu| = \frac 1 2 t + C$.

Last edited: Feb 23, 2016
4. Feb 25, 2016

### Simon Bridge

It's the way that produces the fewer headaches.

5. Feb 25, 2016

### Staff: Mentor

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.

6. Feb 28, 2016

### Simon Bridge

I was unclear - I was agreeing with you.