anonymous
  • anonymous
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.
Mathematics
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SOLVED
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jamiebookeater
  • jamiebookeater
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jim_thompson5910
  • jim_thompson5910
Is the given DE this? \[\Large \frac{dy}{dx} = 6x + \frac{9}{x}\] OR is it this? \[\Large \frac{dy}{dx} = \frac{6x + 9}{x}\]
anonymous
  • anonymous
It's exactly like that first one.
anonymous
  • anonymous
where did ya get the log from lol

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jim_thompson5910
  • jim_thompson5910
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
jim_thompson5910
  • jim_thompson5910
@magepker728 \[\Large \int \frac{1}{x}dx = \ln(x)+C\]
anonymous
  • anonymous
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.
jim_thompson5910
  • jim_thompson5910
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
jim_thompson5910
  • jim_thompson5910
sorry, I meant "restrict the argument of the natural log to be positive only"
anonymous
  • anonymous
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?
jim_thompson5910
  • jim_thompson5910
yeah I'd answer it as \[\Large 3x^2 + 9\ln(|x|) + C\] I don't think there's anything more
anonymous
  • anonymous
That was the correct answer. I guess I was pretty close, but I had forgotten about absolute value. Thank you for the help.
jim_thompson5910
  • jim_thompson5910
no problem

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