Question

So, I'm doing some homework and the first problem says:
Find the general solution of the differential equation. Use C for the constant of integration.
dy/dx = 6x + 9/x

I thought the answer was 3x^2 + 9log(x) + C, but the website says it is wrong.

Answer 1

Is the given DE this?
\[\Large \frac{dy}{dx} = 6x + \frac{9}{x}\]
OR is it this?
\[\Large \frac{dy}{dx} = \frac{6x + 9}{x}\]

Answer 2

It's exactly like that first one.

Answer 3

where did ya get the log from lol

Answer 4

you're very close. It's not a log of base 10, it's going to be a log of base 'e'

so you either say \(\Large \log_e(x)\) or you say \(\Large \ln(x)\)

the second way is much more efficient

Answer 5

@magepker728
\[\Large \int \frac{1}{x}dx = \ln(x)+C\]

Answer 6

I tried ln(x) in my answer as well for my second attempt and it said that was wrong, too. I looked at the book and I'm pretty sure I have the right idea, but maybe I'm messing up that 9/x part of the problem.

Answer 7

well keep in mind that the domain of ln(x) is x > 0

so maybe they want you to restrict x to be positive only. One way to do that is to use absolute value

ie

\[\Large \int \frac{1}{x}dx = \ln(|x|)+C\]
where x is nonzero

Answer 8

sorry, I meant "restrict the argument of the natural log to be positive only"

Answer 9

Oh, I hadn't thought of absolute value, but that makes sense. So I would answer with \[\ln(\left| x \right|)+C\]or is there more to the answer than this?

Answer 10

yeah I'd answer it as
\[\Large 3x^2 + 9\ln(|x|) + C\]
I don't think there's anything more

Answer 11

That was the correct answer. I guess I was pretty close, but I had forgotten about absolute value. Thank you for the help.

Answer 12

no problem