Integrating Factor, how do you get this?

[flyingpig]

Homework Statement

My book shows that

[PLAIN]http://img845.imageshack.us/img845/4875/unleduot.jpg

and then they arrived at the results

[PLAIN]http://img202.imageshack.us/img202/5442/unleddq.jpg

So my problem is that, how exactly did they drop the mu in the first picture?

 

[HallsofIvy]

I don't understand your question. I can see nowhere that the "dropped" the mu. In the "first picture", they wound up solving for mu.

 

[flyingpig]

[PLAIN]http://img703.imageshack.us/img703/2056/unledid.jpg

 

[HallsofIvy]

As I said, they didn't "drop" mu, they solved for it!
If
$$\frac{du}{dx}= F(x)u$$
then
$$\frac{du}{u}= F(x)dx$$
and, integrating both sides,
$$ln(u)= \int F(x)dx$$

Now take the exponential of both sides.

 

[flyingpig]

Ohhh okay that wasn't clear to me at first. Is there any good way to memorize it? It takes too long to derive it lol

 

[HallsofIvy]

The fact that
$$\frac{dy}{dt}= f(x)g(y)$$
gives
$$\frac{dy}{g(y)}= f(x)dx$$
should be immediate from Calculus.

 

[flyingpig]

How do know that mu(x) is linear?