Question

The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.

Options

• True

• False

Answer 1

This statement is True.

Explanation:

The given differential equation is `("d"y)/("d"x) = (y/x)^(1/3)`

⇒ `("d"y)/("d"x) = y^(1/3)/x^(1/3)`

⇒ `("d"y)/y^(1/3) = ("d"x)/x^(1/3)`

Integrating both sides, we get

`int ("d"y)/y^(1/3) = int ("d"x)/x^(1/3)`

⇒ `int y^(-1/3) "d"y = int x^(-1/3) "d"x`

⇒ `1/(- 1/3 + 1) y^(-1/3 + 1) = 1/(-1/3 + 1) * x^(-1/3) "d"x`

⇒ `1/(- 1/3 + 1) y^(-1/3 + 1) = 1/(-1/3 + 1) * x^(-1/3 + 1) + "c"`

⇒ `3/2 y^(2/3) = 3/2 x^(2/3) + "c"`

⇒ `y^(2/3) = x^(2/3) + 2/3 "c"`

⇒ `y^(2/3) - x^(2/3) = "k"["k" = 2/3 "c"]`