Question

Solve the differential equation:

`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1

Answer 1

`e^(x/y) (1- x/y) + (1 + e^(x/y)) dx/dy = 0`

`Put x = vy

`dx/dy = v + y "dv"/dy`

`:. e^v (1 - v) + (1 + e^v).(v + y "dv"/dy) = 0`

`v(1 + e^v) + y(1 + e^v). (dv)/dy = (v - 1)e^v`

`y(1 + e^v) (dv)/dy = e^v v - e^v - ve^v - v`

`y(1 + e^v) (dv)/(dy) = - (v + e^v)` 

`(1 + e^v)/(-(v + e^v)) "dv"/dy = 1/y`

`-int (1 + e^v)/(v + e^v) dv = int 1/y dy`

`log c - log(v + e^v) = log y`

`c/(v + e^v) = y`

`y(v + e^v) = c`

`c = y(v + e^v)` 

`c = y(x/y + e^(x"/"y))`     ....(1)

when x = 0, y = 1

`c = 1(0 + e^o)`

c = 1

Put c = 1 in equation (1)

`1 = y(x/y + e^(x/y))`