Question

The general solution of x `"dy"/"dx" = "y" - x tan ("y"/x)`is ________.

पर्याय

• `x^2 sin(x/y)` = c

• `x sin(x/y)` = c

• `x sin(y/x)` = c

• `x^2 sin(y/x)` = c

Answer 1

The general solution of x `"dy"/"dx" = "y" - x tan ("y"/x)`is `underline(x sin(y/x) = "c")`.

Explanation:

We have, x`"dy"/"dx" = y - x tan (y/x)`

`=> "dy"/"dx" = y/x - tan(y/x)`

Put `y/x = "v" => "y" = "vx"`

`=> "dy"/"dx" = "v" + x "dv"/"dx"`

`therefore "v + x" "dv"/"dx" = "v" - tan "v"`

`=> x "dv"/"dx" = - tan "v"`

⇒ cot v dv = - `"dx"/x`

⇒ log(sin v) = - log(x) + log c

⇒ log (sin v · x) = log c

⇒ sin`(y/x)*x` = c

⇒ x sin `(y/x) = "c"`