Diff. Eqn: y(lnx-lny)dx = (xlnx-xlny-y)

[pr0blumz]

y(lnx-lny)dx = (xlnx-xlny-y)

Homework Statement

Homework Equations

The Attempt at a Solution

dy/dx = y(lnx-lny)/(xlnx-xlny-y)

y = ux

dy/dx = u + xu'

u + xu' = ux(lnx-lnux)/(xlnx-xlnux-ux)

u + xu' = (ulnx-ulnux-ulnux)/(lnx-lnux-u)

xu'= (ulnx-ulnux-ulnx)/(lnx-lnux-u)-u

xu'= (ulnx-ulnux-ulnx+ulnux+u^2)/(lnx-lnux-u)

xu' = (u^2)/(lnx-lnux-u)

(lnx-lnux-u) /(u^2) *du = dx/x

(lnx-lnux-u) /(u^2)* du - dx/x = 0

Am I correct so far or did I mess up somewhere? Thanks

Edit: I figured out that I had to solve for dx/dy since M(x,n) was simpler and solved the problem.

 

[mighty2000]

Yes, you are correct so far. You have correctly applied the separation of variables method to solve the differential equation. However, as you mentioned, you will need to solve for dx/dy to find the final solution. Keep in mind that the solution you have obtained is in implicit form, so you will need to solve for y in terms of x to get the explicit solution.