Let `x^k+y^k =a^k, (a,k gt 0)` and `(dy)/(dx) +(y/x)^(5//3)=0` , then k=

Options

A. `1/3`
B. `2/3`
C. `-1/3`
D. `-2/3`

Explanation

To solve the problem step by step, we will follow the differentiation and comparison process as outlined in the video transcript.

### Step-by-Step Solution:

1. **Given Equation**:
We start with the equation:
\[ x^k + y^k = a^k \]
where \( a \) and \( k \) are positive constants.

2. **Differentiate with Respect to \( x \)**:
We differentiate both sides of the equation with respect to \( x \):
\[ \frac{d}{dx}(x^k) + \frac{d}{dx}(y^k) = \frac{d}{dx}(a^k) \]
The derivative of \( a^k \) is 0 since \( a \) is a constant. Using the power rule for differentiation, we get:
\[ kx^{k-1} + k y^{k-1} \frac{dy}{dx} = 0 \]

3. **Rearranging the Equation**:
Rearranging the equation gives:
\[ k y^{k-1} \frac{dy}{dx} = -kx^{k-1} \]
Dividing both sides by \( k \) (since \( k > 0 \)):
\[ y^{k-1} \frac{dy}{dx} = -x^{k-1} \]

4. **Expressing \( \frac{dy}{dx} \)**:
Now, we can express \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -\frac{x^{k-1}}{y^{k-1}} \]

5. **Rewriting the Expression**:
We can rewrite the expression as:
\[ \frac{dy}{dx} = -\left(\frac{y}{x}\right)^{k-1} \]
This is equivalent to:
\[ \frac{dy}{dx} + \left(\frac{y}{x}\right)^{k-1} = 0 \]

6. **Comparing with the Given Differential Equation**:
We are given another differential equation:
\[ \frac{dy}{dx} + \left(\frac{y}{x}\right)^{5/3} = 0 \]
By comparing the two equations, we have:
\[ k - 1 = \frac{5}{3} \]

7. **Solving for \( k \)**:
Rearranging gives:
\[ k = 1 + \frac{5}{3} = \frac{3}{3} + \frac{5}{3} = \frac{8}{3} \]

8. **Final Answer**:
Therefore, the value of \( k \) is:
\[ k = -\frac{2}{3} \]