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Question
Solve the following differential equation:
`y"e"^(x/y) "d"x = (x"e"^(x/y) + y) "d"y`
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Solution
The given equation can be written as
`("d"x)/("d"y) = (x"e"^(x/y) + y)/(y"e"^(x/y))` ........(1)
This is a Homogeneous differential equation
Put x = vy
⇒ `("d"x)/("d"y) = "v" + y * "dv"/("d"y)`
(1) ⇒ `"v" + y * "dv"/("d"y) = ("vve"^"v" + y)/(y"e"^"v")`
`"v" + y * "dv"/("d"y) = (y("ve"^"v" + 1))/(y"e"^"v")`
`y "d"/("d"y) = ("ve"^"v" + 1)/"e"^"v" - "v"`
`y "dv"/("d"y) = ("ve"^"v" + 1 - "ve"^"v")/"e"^"v"`
`y "dv"/("d"y) = 1/"e"^"v"`
Seperating the variables
`int "e"^"v" "dv" = int ("d"y)/y`
ev = log y + log c
ev = log |cy|
i.e., `"e"^(x/y)` = log |cy| .......`[∵ "v" = x/y]`
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