Maharashtra State BoardHSC Arts 12th Board Exam
Advertisement Remove all ads

Find the equation of the curve which passes through the origin and has the slope x + 3y - 1 at any point (x, y) on it. - Mathematics and Statistics

Sum

Find the equation of the curve which passes through the origin and has the slope x + 3y - 1 at any point (x, y) on it.

Advertisement Remove all ads

Solution

Let A (x, y) be the point on the curve y = f(x).

Then slope of the tangent to the curve at point A is `"dy"/"dx"`.

According to the given condition,

`"dy"/"dx" = "x" + 3"y" - 1`

∴ `"dy"/"dx" - 3"y" = "x - 1"`      ....(1)

This is the linear differential equation of the form

`"dy"/"dx" + "Py" = "Q"`, where P = - 3 and Q = x - 1

∴ I.F. = `"e"^(int "P dx") = "e"^(int - 3"dx") = "e"^(- 3"x")`

∴ the solution of (1) is given by

`"y" * ("I.F.") = int "Q" * ("I.F.") "dx" + "c"`

∴ `"y" * "e"^(- 3"x") = int ("x - 1") * "e"^(-3"x") "dx" + "c"`

∴ `"e"^(- 3"x") * "y" = ("x - 1") int "e"^(- 3"x") - int ["d"/"dx" ("x - 1") * int "e"^(- 3"x") "dx"] "dx" + "c"_1`

∴ `"e"^(- 3"x") * "y" = ("x - 1") * "e"^(- 3"x")/-3 - int 1 * "e"^(- 3"x") * "y"/-3 "dx" + "c"_1`

∴ `"e"^(- 3"x") * "y" = - 1/3 ("x - 1") * "e"^(- 3"x") + 1/3 int "e"^(- 3"x") "dx" + "c"_1`

∴ `"e"^(- 3"x") * "y" = - 1/3 ("x - 1") "e"^(- 3"x") + 1/3 * "e"^(- 3"x")/-3 + "c"_1`

∴ `"e"^(- 3"x") * "y" = - 1/3 ("x - 1")"e"^(- 3"x") - 1/9 "e"^(- 3"x") + "c"_1`

∴ 9y = - 3(x - 1) - 1 + 9`"c"_1 * e^("3x")`

∴ 9y + 3(x - 1) + 1 = `9"c"_1 * e^("3x")`

∴ 9y + 3x - 3 + 1 = `9"c"_1 * e^("3x")`

∴ 3(x + 3y) = 2 + `9"c"_1 * e^("3x")`

∴ 3(x + 3y) = 2 + `"c" * e^("3x")` where c = 9c1   ....(2)

This is the general equation of the curve.

But the required curve is passing through the origin (0, 0).

∴ by putting x = 0 and y= 0 in (2), we get

0 = 2 + c

∴ c = - 2

∴ from (2), the equation of the required curve is

3(x + 3y) = 2 - 2`e^("3x")`

i.e. 3(x + 3y) = 2(1 - `e^("3x")`).

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×