Question

If a curve y = f(x) passes through the point (1, - 1) and satisfies the differential equation, y (1 + xy) dx = x dy, then `f(-1/2)` is equal to ______ 

Options

• `-4/5`

• `2/5`

• `4/5`

• `-2/5`

Answer 1

If a curve y = f(x) passes through the point (1, - 1) and satisfies the differential equation, y (1 + xy) dx = x dy, then `f(-1/2)` is equal to `underline(4/5)`.

Explanation:

y(1 + xy) dx = x dy

⇒ `(ydx - xdy)/y^2 = -x dx`

⇒ `d(x/y) = -x dx`

Integrating on both sides, we get

`x/y = (-x^2)/2 + c` ................(i)

Since the curve passes through (1, -1),

`-1 = (-1)/2 + c ⇒ c = (-1)/2`

∴ `x/y = (-x^2)/2 - 1/2` ........[From (i)]

⇒ `y = (-2x)/(x^2 + 1)`

i.e., f(x) = `(-2x)/(x^2 + 1)`

∴ `f(-1/2) = 4/5`