1. ## linear differential equations??

I have a few questions about finding the general solution of first-order linear differential equations..

http://i28.tinypic.com/2yv3343.gif

http://i31.tinypic.com/2818w02.gif

http://i28.tinypic.com/n81ee.gif

2. Originally Posted by iloveyou33
I have a few questions about finding the general solution of first-order linear differential equations..

http://i28.tinypic.com/2yv3343.gif

http://i31.tinypic.com/2818w02.gif

http://i28.tinypic.com/n81ee.gif
the integrating factor method takes care of all these. see post #21 here

3. it doesn't make sense though..
is there a simpler way of explaining it?
I don't understand what "the integrating factor" is in the first place..

4. Originally Posted by iloveyou33
it doesn't make sense though..
is there a simpler way of explaining it?
I don't understand what "the integrating factor" is in the first place..
it is just something you multiply through by to make the left hand side the derivative given by the product and the integrating factor. for the first one:

$\displaystyle y' + 2y = 3e^t$

(with practice, you will be able to recognize the integrating factor immediately, but let's go through the method to see it)

the integrating factor is $\displaystyle e^{\int 2~dt} = e^{2t}$

multiply through by the integrating factor, we get:

$\displaystyle e^{2t}y' + 2e^{2t}y = 3e^{3t}$

now the left hand side is the result of differentiating $\displaystyle e^{2t}y$ by the product rule, thus

$\displaystyle (e^{2t}y)' = 3e^{3t}$

integrate both sides, we get:

$\displaystyle e^{2t}y = e^{3t} + C$

$\displaystyle \Rightarrow y = e^t + Ce^{-2t}$

the others are done similarly

5. Originally Posted by Jhevon
the integrating factor method takes care of all these. see post #21 here
You always give that example, I think you may put a link in your signature with that.

6. Originally Posted by Krizalid
You always give that example, I think you may put a link in your signature with that.
Haha, yeah. It's kinda weird though. I do not want something in my signature as particular as a kind of differential equations problem. I want general things in my signature, like yours!