Question

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

Answer 1

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

∴`x/y + 2y^2 = dx/dy`

∴ `dx/dy - x/y = 2y^2`

The given equation is of the form

`dx/dy + px =Q`, where, `P = -1/ y and Q = 2y^2`

∴ I.F. `= e ^(int^(pdy)  = e^ (-int^(1/y dy)`

`= e ^(-logy) = e^(1/y)`

`=1/y`

∴ Solution of the given equation is

`x(I.F.) = intQ(I.F.) dy + c`

∴ `x/y =2 int y^2/y d y +c `

∴ `x/y= 2 int y  dy +c `

∴ `x/y= 2  y^2/2 +c `

∴ x = y(c + y²)