Hi, I have a differential equation that I'm trying to solve my substitution. I've got most of it, but I can figure out how to end it properly.
The equation: x y' - y = y [1 - ln2(y/x)]
I divide by x
let v = y/x
y' = v + xv'
sub v and y' in the equation to get:
dx/x = dv/[v(1-ln2v)]
let u = ln v
du = dv/v
sub these to get:
dx/x = du/(1-u2)
Integrating this I get:
ln |x| = 1/2 * ln | (1+u)/(1-u) |
reverse substituting v:
ln |x| = 1/2 * ln | (1+ ln v) / (1 - ln v ) |
reverse substituting y:
ln |x| = 1/2 * ln | [1+ ln (y/x)] / [1- ln (y/x)] |
How do I proceed after this. Is there any way to get the solution in terms of y?
Or should I use a different approach from the beginning itself?
Any help is appreciated, thanks