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

Options

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

Answer

To find the degree of the given differential equation

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

we will follow these steps:

### Step 1: Identify the highest order derivative
The highest order derivative in the equation is \(\frac{d^2 y}{dx^2}\), which is a second-order derivative.

### Step 2: Rewrite the equation in standard form
The equation can be rewritten as:

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

### Step 3: Determine the polynomial form
To find the degree, we need to express the equation as a polynomial in terms of the highest order derivative. Here, we can see that the term \(2 \left(\frac{d^2 y}{dx^2}\right)^2\) is a polynomial of degree 2 in \(\frac{d^2 y}{dx^2}\).

### Step 4: Identify the degree of the polynomial
The degree of a differential equation is defined as the highest power of the highest order derivative when the equation is expressed as a polynomial. In this case, the highest order derivative is \(\frac{d^2 y}{dx^2}\) and it appears squared in the equation.

### Conclusion
Thus, the degree of the given differential equation is **2**.

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