The degree of the differential equation
`(d^(2)y)/(dx^(2))+3((dy)/(dx))^(2)=x^(2)log((d^(2)y)/(dx^(2))),` is

Options

A. 1
B. 2
C. 3
D. none of these

Explanation

To find the degree of the given differential equation:

\[ \frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right)^2 = x^2 \log\left(\frac{d^2y}{dx^2}\right) \]

we need to analyze the equation step by step.

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

**Hint:** Look for the term with the highest derivative in the equation to determine the order.

### Step 2: Check if the equation is polynomial in derivatives
A differential equation is said to be polynomial in its derivatives if all the derivatives can be expressed as a polynomial (i.e., no transcendental functions like logarithms or exponentials are involved).

In our equation, we see that the term \(\log\left(\frac{d^2y}{dx^2}\right)\) involves a logarithm of the second derivative. This means that the equation is not polynomial in its derivatives.

**Hint:** Evaluate each term in the equation to see if they can be expressed as polynomials.

### Step 3: Determine the degree
Since the equation involves a logarithmic function of the highest order derivative, the degree of the differential equation is not defined.

**Hint:** If any term in the equation is not polynomial in the derivatives, the degree is considered undefined.

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

**Final Answer:** The degree of the differential equation is undefined.