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Question
Solve the following differential equation:
`[x + y cos(y/x)] "d"x = x cos(y/x) "d"y`
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Solution
The given equation can be written as
`("d"y)/("d"x) = (x + y cos(y/x))/(x cos y/x)` ........(1)
This is a homogeneous differential equations.
Put y = vx
`("d"y)/("d"x) = "v" + x "dv"/("d"x)`
1 ⇒ ∴ `"v" + x "dv"/("d"x) = (x + "v"x cos((vx)/x))/(x cos((vx)/x))`
`"v" + x "dv"/("d"x) = (x[1 + "v" cos "v"])/cos "v"`
`x "dv"/("d"x) = (1 + "v" cos "v")/cos"v" - "v"`
`x "dv"/("d"x) = 1/cos"v"`
cos v dv = `("d"x)/x`
On integration we obtain
`int cos"v" "d"v = int ("d"x)/x`
sin v = log x + log c
`sin(y/x)` = log x + log c
`sin(y/x)` = log |cx|
Which gives the required solution.
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