Question

Solve the differential equation by separation of variables ylnx(dx/dy)=((y+1)/x)^2

Answer 1

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.

Answer 2

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).

Answer 3

I understand the concept but when I actually try to solve it, I get stuck.

Answer 4

How far did you get?

Answer 5

xylnx(dx)=(y+1)^2 dy and I'm not even sure if that is correct

Answer 6

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.

Answer 7

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

Answer 8

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.

Answer 9

I can check your work on the differential equation later if you want to.

Answer 10

ok thank you very much that actually makes sense know, I had just forgotten about that technique!!

Answer 11

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.

Answer 12

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.

Answer 13

Yeah, you're right, that's way better.