Question

Need help with differential equations :/.
Find a second solution for xy''+(x-1)y'-y=0, given that y1=e^-x.

Answer 1

Do you know how to use integrating factor?

Answer 2

Oh... yes I do but I thought that only worked if the equation was in standard form and a first order equation. y'+P(x)y=q(x)

Answer 3

In that case I would just multiply everything by e^(integral (p) dx). Does that work on this too?

Answer 4

Ah wait do I need to do reduction of order to get it in the right form?

Answer 5

Oh wait, give me a few minutes. I'm not sure if it applies here.

Answer 6

okay thank you. I appreciate it because I am definitely lost :/.

Answer 7

I actually am taking Differential Equations right now for my engineering major, but we haven't covered second order differential equations with nonconstant coefficients. I would recommend however visiting this page http://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx
It seems to be close to what you are looking for. :)

Answer 8

Okay I will read over it right now. Thank you for the link :). and that is interesting. I studied molecular biology but I am switching to biosystems engineering.

Answer 9

If they were constants, I'd totally be able to help you using the different cases :)

I can't do anything biology for the life of me, haha. I admit I find it extremely fascinating but I can never memorize all the terms and functions XD

Answer 10

hmm... okay so I did find an answer key to the question but I am still not sure how the professor solved this. Apparently the answer is y2=x-1 but I am still trying to figure out what she did.

Answer 11

I think I understand the basic idea a solution to the differential equation can be a scalar multiple of the given y1. but I don't understand where the integrating factor comes in... this is so frustrating :/.

Answer 12

I don't understand why he used integrating factor at first. It seems like a different method than reduction of order, but I actually understand what he's doing.

It seems like after finding e^-u, he integrating in the following form
\[y_2=y_1* \int\limits \frac{ e^{-\mu} }{ y_1{^2} }dx\]

Then once he simplified that, he just plugged in y1 that was given from the beginning. Also note that at the end, he put an e^-x instead of e^x for the last term in the parentheses. Or vice versa. Not sure which one it's suppose to be but there's an error there.

Answer 13

Does this make sense now?

Answer 14

I sort of understand but I don't really get why the professor is multiplying by ∫e^(−μ)/y^2)dx. Where does that come from?

Answer 15

I'm assuming it's some sort of formula to follow. I wouldn't really know because I haven't learned this yet (until now though, so thank you :P). But I don't understand the logic behind it and where it was derived from. For now I would just go with it and ask your professor later for clarification.

Answer 16

haha... okay... well, if it works on the exam, I won't question it lol.

Answer 17

:)

Answer 18

Thanks for the help :). Good luck with your studies as well. Hopefully we can help eachother out :p

Answer 19

You are very welcome! And hopefully :P