If the independent variable `x` is changed to `y ,` then the differential equation `x(d^2y)/(dx^2)+((dy)/(dx))^3-(dy)/(dx)=0` is changed to `x(d^2x)/(dy^2)+((dx)/(dy))^2=k` where `k` equals____

Explanation

To solve the problem, we need to transform the given differential equation with respect to the variable \( x \) into one with respect to the variable \( y \).

### Step-by-Step Solution:

1. **Given Differential Equation**:
The original differential equation is:
\[ x \frac{d^2y}{dx^2} + \left( \frac{dy}{dx} \right)^3 - \frac{dy}{dx} = 0 \]

2. **Change of Variables**:
We need to change the independent variable from \( x \) to \( y \). We know that:
\[ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} \]

3. **Finding \( \frac{d^2y}{dx^2} \)**:
To find \( \frac{d^2y}{dx^2} \) in terms of \( y \), we use the chain rule:
\[ \frac{d^2y}{dx^2} = \frac{d}{dy} \left( \frac{dy}{dx} \right) \cdot \frac{dy}{dx} \]
Hence, we can express \( \frac{d^2y}{dx^2} \) as:
\[ \frac{d^2y}{dx^2} = \frac{d}{dy} \left( \frac{dy}{dx} \right) \cdot \frac{dy}{dx} = \frac{d^2y}{dy^2} \cdot \left( \frac{dy}{dx} \right) \]

4. **Substituting into the Equation**:
Substitute \( \frac{d^2y}{dx^2} \) into the original equation:
\[ x \left( \frac{d^2y}{dy^2} \cdot \left( \frac{dy}{dx} \right) \right) + \left( \frac{dy}{dx} \right)^3 - \frac{dy}{dx} = 0 \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ x \frac{d^2y}{dy^2} \left( \frac{dy}{dx} \right) + \left( \frac{dy}{dx} \right)^3 - \frac{dy}{dx} = 0 \]
This can be rewritten as:
\[ x \frac{d^2y}{dy^2} \left( \frac{dy}{dx} \right) = \frac{dy}{dx} - \left( \frac{dy}{dx} \right)^3 \]

6. **Finding \( k \)**:
We can express the equation in the form:
\[ x \frac{d^2x}{dy^2} + \left( \frac{dx}{dy} \right)^2 = k \]
By comparing terms, we find that:
\[ k = 1 \]

### Final Answer:
Thus, the value of \( k \) is:
\[ \boxed{1} \]