what is the degree of the differential equation ` (d^(3)y)/(dx^(3) +2 ((d^(2)y)/(dx^(2)))^(2) - (dy)/(dx) =y=0` ?

Options

A. 6
B. 3
C. 2
D. 1

Explanation

To find the degree of the given differential equation

\[ \frac{d^3y}{dx^3} + 2 \left( \frac{d^2y}{dx^2} \right)^2 - \frac{dy}{dx} + y = 0, \]

we will follow these steps:

### Step 1: Identify the highest order of the derivative
The first step is to identify the highest order of the derivative present in the equation.

In our equation, we have:
- \(\frac{d^3y}{dx^3}\) (third derivative)
- \(\frac{d^2y}{dx^2}\) (second derivative)
- \(\frac{dy}{dx}\) (first derivative)

The highest order of the derivative is \(3\) (from \(\frac{d^3y}{dx^3}\)).

### Step 2: Determine the degree of the differential equation
The degree of a differential equation is defined as the power of the highest order derivative when the equation is a polynomial in derivatives.

In our case, the highest order derivative is \(\frac{d^3y}{dx^3}\), and it appears to the power of \(1\).

### Conclusion
Thus, the degree of the differential equation is \(1\).

### Final Answer
The degree of the differential equation is \(1\).

---