Order and degree of the differential equation `|{:(x^(3),,y^(2),,3),(2x^(2),,3y(dy)/(dx),,0),(5x,,2(y(d^(2)y)/(dx^(2))+((dy)/(dx))^(2)),,0):}|` =0 respectively are

Options

A. 1 and 1
B. 1 and 2
C. 2 and 2
D. 2 and 1

Answer

To find the order and degree of the given differential equation represented by the determinant:

\[ \begin{vmatrix} x^3 & y^2 & 3 \\ 2x^2 & 3y & 0 \\ 5x & 2\left(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right) & 0 \end{vmatrix} = 0 \]

we will follow these steps:

### Step 1: Write down the determinant
We start with the determinant of the matrix set to zero:

\[ \begin{vmatrix} x^3 & y^2 & 3 \\ 2x^2 & 3y & 0 \\ 5x & 2\left(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right) & 0 \end{vmatrix} = 0 \]

### Step 2: Expand the determinant
We will expand the determinant using the third column:

\[ = 3 \cdot \begin{vmatrix} 2x^2 & 3y \\ 5x & 2\left(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right) \end{vmatrix} - 0 + 0 \]

Calculating the 2x2 determinant:

\[ = 3 \left(2x^2 \cdot 2\left(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right) - 5x \cdot 3y\right) \]

### Step 3: Simplify the expression
Simplifying the expression gives:

\[ = 3 \left(4x^2\left(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right) - 15xy\right) = 0 \]

### Step 4: Rearranging the equation
We can rearrange this to:

\[ 4xy\left(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right) - 15xy = 0 \]

Factoring out \(xy\):

\[ xy\left(4\left(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right) - 15\right) = 0 \]

### Step 5: Identify the order and degree
Now, we can identify the order and degree of the differential equation:

- **Order**: The highest derivative present in the equation is \(\frac{d^2y}{dx^2}\), which indicates that the order is **2**.

- **Degree**: The degree is determined by the power of the highest order derivative when the equation is a polynomial in derivatives. Here, \(\frac{d^2y}{dx^2}\) appears to the power of 1, so the degree is **1**.

### Final Answer
Thus, the order and degree of the given differential equation are:

- **Order**: 2
- **Degree**: 1

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