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

Options

A. `x=y^(3) +Cy`
B. `x = y^(3)+2Cy`
C. ` x = 2y^(3)+Cy`
D. ` x=3y^(3)+Cy`

Answer

a

Explanation

The Given differential equation can be written as
` (dx)/(dy ) + (-1/y) x = 2y^(2) , y != 0 `
This is a linear differential equation of the form
` (dx)/(dy) +Px =Q `
Here, `P = (-1)/y and Q = 2y^(2)`
` :. IF = e^( intPdy ) = e^(int (-1)/y dy) = e^(-logy ) = y^(-1) = y^(-1) = 1/y `
The general solution of the given differential equation is given by
` x IF = int (Q xx IF) dy + C`
` rArr x(1/y) = int 2y^(2) *1/y dy +C`
` rArr x/y = y^(2) +C rArr x = y^(3) + Cy`