Question

The solution of the differential equation y² dx + (x² − xy + y²)dy = 0 is ______.

विकल्प

• `tan^-1 (x/y) + ln y + C = 0`

• `2 tan^-1 (x/y) + ln x + C`

• `ln(v + sqrt(x^2 + y^2)) + ln y + C = 0`

• `ln (x + sqrt(x^2 + y^2)) + C = 0`

Answer 1

The solution of the differential equation y² dx + (x² − xy + y²)dy = 0 is `bbunderline(tan^-1 (x/y) + ln  y + C = 0)`.

Explanation:

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

Put x = vy

⇒ `(dx)/(dy) = v = y (dv)/(dy)`

∴ `v + y (dv)/(dy) + v^2 - v + 1 = 0`

⇒ `-y (dv)/(dy) = (1 + v^2)`

⇒ `(dv)/(v^2 + 1) = -(dy)/y`

Integrating, we get

tan⁻¹ v + C = −In y

⇒ `tan^-1 (x/y) + ln y + C = 0`

where C is arbitrary constant