Question

The integrating factor of differentital equation `(x + 2y^3) (dy)/(dx) = 2y` is ______.

Options

• `e^(y^2/2)`

• `1/sqrty`

• `1/y^2`

• `e^(-1/y^2)`

Answer 1

The integrating factor of differentital equation `(x + 2y^3) (dy)/(dx) = 2y` is `bbunderline(1/sqrty)`.

Explanation:

`(x + 2y^3) (dy)/(dx) = 2y`

`(dy)/(dx) = (2y)/(x + 2y^3)`

`(dx)/(dy) = (x + 2y^3)/(2y)`

`(dx)/(dy) = (x)/(2y) + (2y^3)/(2y)`

`(dx)/(dy) = (x)/(2y) + y^2`

`(dx)/(dy) - (x)/(2y) = y^2`   ...(Linear differential equation)

Here, `P = (-1)/(2y), θ = y^2`

`I.F. = e^(intP  dy)`

`= e^(-int 1/(2y)  dy)`

`= e^((-1)/2 log y)`

`= e^(log  1/sqrty)`

`= 1/sqrty`