If m be the order and n the degree of the differential equation `((d^(2)y)/(dx^(2)))^(2)+x((dy)/(dx))^(2)=0` then (m, n) =

Options

A. (2, 2)
B. (3, 3)
C. (2, 3)
D. none

Answer

To find the order \( m \) and degree \( n \) of the given differential equation

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

we will follow these steps:

### Step 1: Identify the Order of the Differential Equation

The order of a differential equation is defined as the highest derivative present in the equation.

In the given equation, we have:
- The term \( \frac{d^2y}{dx^2} \) which is the second derivative of \( y \).
- The term \( \frac{dy}{dx} \) which is the first derivative of \( y \).

The highest derivative present is \( \frac{d^2y}{dx^2} \), which is the second derivative. Therefore, the order \( m \) of the differential equation is:

\[ m = 2. \]

### Step 2: Identify 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 equation, we can rewrite it as:

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

Here, the highest order derivative \( \frac{d^2y}{dx^2} \) is raised to the power of 2. Thus, the degree \( n \) of the differential equation is:

\[ n = 2. \]

### Step 3: Conclusion

Now that we have determined both the order and the degree, we can express the final answer as:

\[ (m, n) = (2, 2). \]