Question

The solution of the differential equation `(y^2 + xy^2)dx/dy + x^2 - yx^2 = 0` is ______ 

पर्याय

• `log(x/y) = 1/x + 1/y + c`

• `log(y/x) = 1/x + 1/y + c`

• log(xy) = `1/x + 1/y + c`

• log(xy) + `1/x + 1/y + c`

Answer 1

The solution of the differential equation `(y^2 + xy^2)dx/dy + x^2 - yx^2 = 0` is `underline(log(x/y) = 1/x + 1/y + c)`.

Explanation:

 `(y^2 + xy^2)dx/dy + x^2 - yx^2 = 0`

⇒ `(x^2 - yx^2)dy/dx + y^2 + xy^2 = 0`

⇒ `x^2(1 - y)dy/dx + y^2(1 + x) = 0`

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

integrating on both sides. we get

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

⇒ `-1/y - logy - 1/x + logx = c`

⇒ `log(x/y) = 1/x + 1/y + c`