Separating variables:

`x\ dy=y\ ln\ y\ dx`

`(dy)/(y\ ln\ y)=(dx)/x`

Integrating: [For the y part, let u = ln y, then du = dy/y].

`int(dy)/(y\ ln\ y)=int(dx)/x`

`ln(ln\ y)=ln\ x+K`

Substituting x = 2 when y = e gives:

`ln(ln\ e)=ln\ 2+K`

`ln(1)=ln\ 2+K`

`0=ln\ 2+K`

`K=-ln\ 2`

Substituting this in our general solution:

`ln(ln\ y)=ln\ x-ln\ 2` `=ln\ x/2`

This gives us:

`ln\ y=x/2`

So the particular solution is given by:

`y=e^(x"/"2)`

Easy to understand math videos:
MathTutorDVD.com