The degree of the differential equation `(dy)/(dx) - x = (y - x"" (dy)/(dx))^(-4)` is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the degree of the given differential equation

\[ \frac{dy}{dx} - x = (y - x \frac{dy}{dx})^{-4} \]

we will follow these steps:

### Step 1: Rearrange the equation
First, we will rearrange the equation to isolate the term on the right side:

\[ \frac{dy}{dx} - x = (y - x \frac{dy}{dx})^{-4} \]

We can rewrite this as:

\[ (y - x \frac{dy}{dx})^{4} = \frac{1}{\frac{dy}{dx} - x} \]

### Step 2: Identify the highest order derivative
In this equation, the highest order derivative is \(\frac{dy}{dx}\).

### Step 3: Determine the degree
The degree of a differential equation is defined as the power of the highest order derivative after the equation has been made polynomial in that derivative.

In our case, we can see that the term \((y - x \frac{dy}{dx})^{-4}\) can be rewritten as:

\[ (y - x \frac{dy}{dx})^{4} = 1 \]

This indicates that the term involving the derivative \(\frac{dy}{dx}\) is raised to the power of 4 when we rearrange it.

### Step 4: Combine the degrees
Now, we have:

- The term \(\frac{dy}{dx}\) itself contributes a degree of 1.
- The term \((y - x \frac{dy}{dx})^{4}\) contributes a degree of 4.

Thus, the total degree of the differential equation is:

\[ 1 + 4 = 5 \]

### Conclusion
Therefore, the degree of the differential equation is **5**.