Hi. I am trying to solve dy/dx= -6y - 8.
-1/6Ln (-6y-8)=x +c.
Dividing by -1/6, you get Ln (-6y-8)= -6x-6c.
I next take the e of both sides to get -6y-8= e^(-6x-6c).
How do I proceed from here, as the correct answer is y= -4/3+ce^-6x?
How are you seperating this equation? Did your instructor say you HAVE to use seperation of variables? The differential is linear:
The thing about these differentials, is that more often than not there are several ways of solving it. Some are way more easier to solve than not.
Actually, I would consider "separating variables" to be easier than finding an integrating factor!
jaijay32, from -6y-8= e^(-6x-6c), you can separate the exponential as -6y- 8= e^(-6x)e^(-6c). Since c is an "unknown constant", e^(-6c) is also some unknown constant so just call it "C": -6y- 8= Ce^(-6x).
(In fact, that is better: the integral of 1/x dx is ln|x|, not ln(x). What you should have got was ln|-6y-8|= ln|6y+8|= -6x- 6c so |6y+ 8|= e^(-6x-6c)= e^(-6x)e^(-6c). The point is that, since an exponential is always positive, "-6y- 8= e^(-6x)e^(-6c)" implies that -6y- 8 is positive. But what you really have is |6y+8|= e^(-6x)e^(-6c) so that 6y+8 can be either positive or negative. Writing "-6y- 8= Ce^(-6x)" or "6y- 8= Ce^(-6x)" includes C being either positive or negative.)
Now, with -6y- 8= Ce^(-6x), -6y= 8+ Ce^(-6x) and then y= -4/3+ (C/-6)e^(-6x)= -4/3+ C'e^(-6x) with C'= C/-6.
If you had written it 6y+ 8= Ce^(-6x), you would have 6y= -8+ Ce^(-6x) and then y= -4/3+ (C/6)e^(-6x) or y= -4/3+ C"e^(-6x), exactly the same answer with C"= C/6 now.