Question

Let the curve y = y(x) be the solution of the differential equation, `("dy")/("d"x) = 2(x + 1)`. If the numerical value of area bounded by the curve y = y(x) and x-axis is `(4sqrt(8))/3`, then the value of y(1) is equal to ______.

Options

• 0

• 1

• 2

• 3

Answer 1

Let the curve y = y(x) be the solution of the differential equation, `("dy")/("d"x) = 2(x + 1)`. If the numerical value of area bounded by the curve y = y(x) and x-axis is `(4sqrt(8))/3`, then the value of y(1) is equal to 2.

Explanation:

Given `("dy")/("d"x) = 2(x + 1)`

⇒ dy = (2x + 2)dx

Now, Integrating above,

`int"dy" = int(2x + 2)"d"x`

⇒ y = `(x^2/2)2 + 2x + "c"`

⇒ y = x² + 2x + c

Area bounded by y = y(x) and x-axis is `(4sqrt(8))/3`

Using equation (i), at x-axis (y = 0)

x² + 2x + c = 0

x = `(-2 +- sqrt(4 - 4"c"))/2`

⇒ x = `-1 +- sqrt(1 - "c")`

A = `2int_(-1)^(-1 + sqrt(1 - "c")){-(x + 1)^2 - "c" + 1}"d"x = (4sqrt(8))/3`

⇒ `[(-(x + 1)^3)/3 - "c"x + x]_(-1)^(-1 + sqrt(1 - "c")) = (2sqrt(8))/3`

⇒ `(-(sqrt(1 - "c"))^3)/3 - "c"(-1 + sqrt(1 - "c")) + (-1 + sqrt(1 - "c")) - "c" + 1 = (2sqrt(8))/3`

⇒ `-(sqrt(1 - "c"))^3 + 3"c" - 3"c"sqrt(1 - "c") -3 + 3sqrt(1 - "c") - 3"c" + 3 = 2sqrt(8)`

⇒ `-(sqrt(1 - "c"))^3 - 3"c"sqrt(1 - "c") + 3sqrt(1 - "c") = 2sqrt(8)`

⇒ c = –1

Now using Equation (i)

y = y(x) = x² + 2x – 1

⇒ y(1) = 2