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Question
The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.
Options
True
False
Solution
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"]`
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