Question

Question #fec67

Answer 1

This is Linear Differential Equation.
`xy= ln(c/y)`

Explanation:

Instead of `dy/dx` we form `dx/dy` because of linear `x` in Numerator.

`y^2dx+(xy+1)dy=0`

Dividing by `dy`

`y^2dx/dy+(xy+1)=0`

Dividing by `y^2`

`dx/dy+(x/y+1/y^2)=0`

`dx/dy +x/y = -1/y^2`

Comparing with:
`dx/dy + P(y).x=Q(y)`

`P(y) = 1/y` and `Q(y) = -1/y^2`

Integrating Factor:
`e^(int 1/y dy)`

`e^lny`

i.e `y`

Now,
`x . IF = int IF . (-1/y^2) dy`

`x y = int y (-1/y^2) dy`

`x y = int -1/y dy`

`xy = -ln y + ln c`

`xy = ln(c/y)`