Question

Solve the following differential equation:

`[x + y cos(y/x)] "d"x = x cos(y/x) "d"y`

Answer 1

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.