The solution of differential equation
` (dy)/(dx) (x^(2) y^(3)+xy) = 1 ` is

Options

A. `1/x = 2-y^(2)+Ce^(-y^(2)//2)`
B. `e^(y^(2)//2)((1-2x)/x-y^(2))=C`
C. ` 2/x = 1-y^(3)+e^(-y^(2)//3)`
D. None of these

Answer

a

Explanation

We can write the given differential equation as
` (dx)/(dy) = x^(2) y^(3) +xy`
` (dx)/(dy) - xy = x^(2) y^(3)`
This is the type of Bernaulli's equation which is to be reducible in linear form .
So , dividing the given equation by `x^(2)`
` rArr x^(-2) (dx)/(dy) - x ^(-1) y = y^(3)`
So put ` x^(-1) = v rArr x^(2) (dx)/(dy) = (-dv)/(dy)`
` :. IF = e^(int ydy ) = e^(y^(2)//2`
Required Solution is ` ve^(y^(2)//2) = - int y^(3) e^(y^(2//2)dy) +C `
` = - ( y^(2) e^(y^(2//2))- 2e^(y^(2)//2)) + C`
` rArr 1/x = (2 - y^(2)) + Ce^(-y^(2//2)`