Question

dy/dx+ y log y/x= y(log)^2/x^2

Answer 1

I assume that logylog⁡y refers to the natural log, and will henceforth write lnyln⁡y.

It is not clear from the way this question is written whether it should be

dxdy+yxlnyordxdy+yxlny=yx2lny=yx2lny(equation 1)(equation 2)dxdy+yxln⁡y=yx2ln⁡y(equation 1)ordxdy+yxln⁡y=yx2ln⁡y(equation 2)

However, in either case, this DE is separable. After separating, we have

x2dx1−x=f(y)dyx2dx1−x=f(y)dy

where f(y)f(y) is either (1) ylnyyln⁡y or (2) ylnyyln⁡y. Integrating the left side gives −12(x2+2x+ln((1−x)2))−12(x2+2x+ln⁡((1−x)2)), while the right side is either

14y2(ln(y2)−1) or Ei(ln(y2))14y2(ln⁡(y2)−1) or Ei⁡(ln⁡(y2))

(plus an arbitrary constant), where Ei(t)=−∫∞−te−ssdsEi⁡(t)=−∫−t∞e−ssds is the exponential integral function.

Neither of these equations is easily amenable to finding an explicit solutions (i.e., solving for either x or y).