Show that the differential equation `2y e^(x/y)\ dx+(y-2x e^(x/y))` `dy=0` is homogeneous. Find the particular solution of this differential equation, given that `x=0` when `y=1.`

Explanation

given differential equation is

`2y e^(x/y)dx+(y-2x e^(x/y))dy=0`
this equation can be written as

`(dx/dy) = (2xe^(x/y)- y )/ (2y e^(x/y))`

`F(x,y) = (2xe^(x/y)- y )/ (2y e^(x/y))......(i) `
this is homogenous equation in the form of `f(x/y)`

put `x= vy`

diff. on both side w.r.t `y`

`(dx)/(dy) = v+ y (dv)/ (dy)`

put in equation (i)

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