Question

If $\sqrt{x} \sqrt[3]{y} = (x + y)^n$ and $x\frac{dy}{dx} - y = 0$, then $n =$

• A. 1
• B. $\frac{6}{5}$
• C. $\frac{5}{6}$
• D. $\frac{4}{9}$

Hint:

Memorize this highly useful shortcut result for competitive exams: If $x^m \cdot y^n = (x+y)^{m+n}$, then it is guaranteed that $\frac{dy}{dx} = \frac{y}{x}$. Recognizing this pattern instantly solves the problem without any differentiation.

Answer 1

The Correct Option is C

To solve this problem, we need to determine the value of \( n \) given the equations:

\(\sqrt{x} \sqrt[3]{y} = (x + y)^n\)

and

\(x\frac{dy}{dx} - y = 0\)

From the second equation, \(x\frac{dy}{dx} - y = 0\), we can rearrange it to express the relationship between \(x\) and \(y\):

• Rewriting the differential equation: \(x\frac{dy}{dx} = y\)
• This implies the relationship: \(\frac{dy}{y} = \frac{dx}{x}\)

Integrating both sides:

• \(\int \frac{dy}{y} = \int \frac{dx}{x}\)
• \(\ln y = \ln x + C\), where \(C\) is the constant of integration.
• Exponentiating both sides gives: \(y = Cx\)

Now, substitute back into the first equation:

• \(\sqrt{x} \sqrt[3]{Cx} = (x + Cx)^n\)
• Simplifying the left side: \(\sqrt{x} (Cx)^{1/3} = C^{1/3} x^{1/2} x^{1/3} = C^{1/3} x^{5/6}\)
• \((1+C)^n x^n\) for the right side

Equate the two expressions since \(x^{5/6} = x^n\):

• \(C^{1/3} x^{5/6} = (1 + C)^n x^n\)

Since the powers of \(x\) must be equal, it follows that:

• \(5/6 = n\)

Therefore, the value of \(n\) is \(\frac{5}{6}\).

The correct answer is: \(\frac{5}{6}\).