Question

Let y = y(x) be the solution of the differential equation `e^xsqrt(1 - y^2)dx + (y/x)dy` = 0, y(1) = –1. Then, the value of (y(3))² is equal to ______.

Options

• 1 + 4e³

• 1 + 4e⁶

• 1 – 4e⁶

• 1 – 4e³

Answer 1

Let y = y(x) be the solution of the differential equation `e^xsqrt(1 - y^2)dx + (y/x)dy` = 0, y(1) = –1. Then, the value of (y(3))² is equal to `underlinebb(1 - 4e^6)`.

Explanation:

Given differential equation `e^xsqrt(1 - y^2)dx + (y/x)dy` = 0, y(1) = –1

⇒ `e^xsqrt(1 - y^2)dx = (-y)/x dy`

⇒ `(ydy)/sqrt(1 - y^2) = -intxe^xdx`

⇒ `int (-ydy)/sqrt(1 - y^2) = intxe^xdx`

⇒ `sqrt(1 - y^2) = e^x(x - 1) + c`

Given x = 1, y = –1

⇒ 0 = 0 + c

⇒ c = 0

⇒ `sqrt(1 - y^2) = e^x(x - 1)`

At x = 3 ⇒ 1 – y² = (e³2)² ⇒ y² = 1 – 4e⁶