Question

The gradient of the curve passing through (4, 0) is given by `"dy"/("d"x) - "y"/x + (5x)/((x + 2)(x - 3))` = 0 if the point (5, a) lies on the curve, then the value of a is ______.

पर्याय

• `67/12`

• `5 sin  7/12`

• `5 log  7/12`

• None of these

Answer 1

The gradient of the curve passing through (4, 0) is given by `"dy"/("d"x) - "y"/x + (5x)/((x + 2)(x - 3))` = 0 if the point (5, a) lies on the curve, then the value of a is `underlinebb(5 log  7/12)`.

Explanation:

The differential equation is

`"dy"/("d"x) - "y"/x = - (5x)/((x + 2)(x - 3))`

I.F = `"e"^(int(-1/x)"d"x) = "e"^(-"ln"  x) = 1/x`

Solution is

`"y"(1/x) = int(1/x) xx (5x)/((x + 2)(x - 3))"d"x = "ln"((x + 2)/(x - 3)) + "C"`

It passes through (4, 0), so C = – ln 6

∴ y = `x  "ln"  {((x + 2))/(6(x - 3))}`

Putting (5, a),

we get a = `5  ln (7/12)`