Question

The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.

पर्याय

• `(2x - 1)/(2y + 3)` = k

• `(y + 1)/(2x - 3)` = k

• `(2x + 3)/(2y - 1)` = k

• `(2x - 1)/(2y - 1)` = k

Answer 1

The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is `(2x + 3)/(2y - 1)` = k.

Explanation:

The given differential equation is (2y – 1)dx – (2x + 3)dy = 0

⇒ (2x + 3)dy = (2y – 1)dx

⇒ `("d"y)/(2y - 1) = ("d"x)/(2x + 3)`

Integrating both sides, we get

`int ("d"y)/(2y - 1) = int ("d"x)/(2x + 3)`

⇒ `1/2 log|2y - 1| = 1/2 log |2x + 3| + log"c"`

⇒ `log|2y - 1| = log|2x + 3| + 2 log "c"`

⇒ `log|2y - 1| - log|2x + 3| = log "c"^2`

⇒ `log|(2y - 1)/(2x + 3)| = log "c"^2`

⇒ `(2y - 1)/(2x + 3) = "c"^2`

⇒ `(2x + 3)/(2y - 1) = 1/"c"^2`

⇒ `(2x + 3)/(2y - 1)` = k

Where k = `1/"c"^2`.