Find the sum of the degree and the order of the differential equation: `y = (x-(2y)/((dy)/(dx)))(((2y)/(dy))/(dx))^(2)`.

Options

A. 2
B. 3
C. 4
D. 6

Answer

To find the sum of the degree and the order of the given differential equation \( y = \frac{x - 2y}{\frac{dy}{dx}} \left( \frac{2y}{\frac{dy}{dx}} \right)^2 \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ y = \frac{x - 2y}{\frac{dy}{dx}} \left( \frac{2y}{\frac{dy}{dx}} \right)^2 \]

### Step 2: Simplify the equation
To simplify, we can take \(\frac{dy}{dx}\) as a common denominator:
\[ y = \frac{(x - 2y) \cdot (2y)^2}{\left(\frac{dy}{dx}\right)^3} \]
This can be rewritten as:
\[ y = \frac{(x - 2y) \cdot 4y^2}{\left(\frac{dy}{dx}\right)^3} \]

### Step 3: Rearranging the equation
Multiplying both sides by \(\left(\frac{dy}{dx}\right)^3\) gives:
\[ y \left(\frac{dy}{dx}\right)^3 = 4y^2 (x - 2y) \]

### Step 4: Expanding and rearranging
Expanding the right-hand side:
\[ y \left(\frac{dy}{dx}\right)^3 = 4xy^2 - 8y^3 \]
Rearranging gives:
\[ 8y^3 - 4xy^2 + y \left(\frac{dy}{dx}\right)^3 = 0 \]

### Step 5: Identify the order and degree
- **Order**: The highest derivative present is \(\frac{dy}{dx}\) and it appears to the power of 3. Since we are differentiating \(y\) only once, the order of the differential equation is 1.

- **Degree**: The degree of the differential equation is the exponent of the highest order derivative when the equation is a polynomial in derivatives. Here, \(\left(\frac{dy}{dx}\right)^3\) indicates that the degree is 3.

### Step 6: Calculate the sum of the order and degree
Now, we can find the sum of the order and degree:
\[ \text{Sum} = \text{Order} + \text{Degree} = 1 + 3 = 4 \]

### Final Answer
The sum of the degree and the order of the differential equation is \(4\).

---