If a and b are arbitrary constants, then the order and degree of the differential equation of the family of curves `ax^(2)+by^(2)=2` respectively are

Options

A. 2, 2
B. 1, 2
C. 1, 1
D. 2, 1

Answer

To find the order and degree of the differential equation of the family of curves given by the equation \( ax^2 + by^2 = 2 \), we will follow these steps:

### Step 1: Differentiate the given equation
Start with the equation:
\[ ax^2 + by^2 = 2 \]

Differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(ax^2) + \frac{d}{dx}(by^2) = \frac{d}{dx}(2) \]

This gives:
\[ 2ax + 2y \frac{dy}{dx} = 0 \]

### Step 2: Rearrange the equation
Rearranging the equation, we get:
\[ 2ax + 2y \frac{dy}{dx} = 0 \implies ax + y \frac{dy}{dx} = 0 \]

### Step 3: Differentiate again
Now, differentiate the equation \( ax + y \frac{dy}{dx} = 0 \) again with respect to \( x \):
\[ \frac{d}{dx}(ax) + \frac{d}{dx}\left(y \frac{dy}{dx}\right) = 0 \]

This gives:
\[ a + \left(\frac{dy}{dx}\right)^2 + y \frac{d^2y}{dx^2} = 0 \]

### Step 4: Identify the order and degree
The highest derivative present in the equation \( a + \left(\frac{dy}{dx}\right)^2 + y \frac{d^2y}{dx^2} = 0 \) is \( \frac{d^2y}{dx^2} \), which indicates that the order of the differential equation is 2.

The degree of the differential equation is determined by the highest power of the highest order derivative. Here, \( \frac{d^2y}{dx^2} \) appears to the first power, so the degree is 1.

### Final Result
Thus, the order and degree of the differential equation are:
- **Order**: 2
- **Degree**: 1