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Seperation of variables means you get the x's and dx on one side and y's and dy on the other side. So it should look like \[ f(x)dx=g(y)dy \] Where \(f(x)\) is something involving x's (no y's!) and \(g(y)\) is something involving y's.
When you have done this, you can integrate both sides and then solve for y, if you can, to get y as a function of x. This is the solution (don't forget integration constants).
I understand the concept but when I actually try to solve it, I get stuck.
How far did you get?
xylnx(dx)=(y+1)^2 dy and I'm not even sure if that is correct
You need to divide by y also, did the differential equation have \(x\) or \(x^2\) in the denominator on the right hand side? If it was \(x^2\) you need another \(x\) on the left hand side.
ok so then I get x^2(lnx)dx=((y+1)^2)/y dy after that I don't know where to go from there since I'm not sure how to integrate each side
You will have to do partial integration or integration by parts. This is a skill you will need to master when solving differential equations, I recommend watching through http://youtu.be/ouYZiIh8Ctc if you need to get repetition on the method or http://youtu.be/LJqNdG6Y2cM which is an example to jog your memory, or to watch after you watch the first one.
I can check your work on the differential equation later if you want to.
ok thank you very much that actually makes sense know, I had just forgotten about that technique!!
For the right hand side you will need to do a substitution to get it into a more familiar form. I would recommend trying something like u=y+1.
Hmm I would expand out the numerator on the right side and then divide y out of each term. :O maybe that's just me though.
Yeah, you're right, that's way better.