An "integrating factor" is a function u(y) such that multiplying both sides by it will make the left side an "exact derivative". That means that you want u(y) such that

d(u(y)x)/dy= u dx/dy+ x du/dy= u dx/dy+ 0.2xu. So you must have du/dy= 0.2u which gives u= e^.2y, not u=e^.2x. That may be a typo or it may be that, because this problem has x as a function of y, rather than the other way around, which is more common, you may have become confused (as I did, initially!). In any case, multiplying both sides of the equation by e^.2y gives e^.2y dx/dy+ e^.2y x= d(e^.2y x)/dy= 3.4y e^(-1.5y)e^.2y= 3.4y e^{-1.3y)

Where in the world did "t" come from? You are integrating "ye^{-1.3y}" which you will need to integrate by "parts".multiply both sides by this and result in xexp^0.5y = 3.4yexp^-t

when i integrate both sides i get as a final answer x = ( 3.4yexp^-y ) / exp^(0.5y)

can anyone please give guidance? method right or way wrong !!

Thanks in advance, and i apologize in advance for the poor formatting above, I could not figure out how you all do it the right way !!