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 determine 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
We start with the equation:
\[ ax^2 + by^2 = 2 \]
We will differentiate this equation with respect to \( x \).

### Step 2: Apply differentiation
Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}(ax^2) + \frac{d}{dx}(by^2) = \frac{d}{dx}(2) \]
Using the product rule on \( by^2 \):
\[ 2ax + b \cdot 2y \cdot \frac{dy}{dx} = 0 \]
This simplifies to:
\[ 2ax + 2by \frac{dy}{dx} = 0 \]

### Step 3: Rearranging the equation
We can rearrange the equation to express \( \frac{dy}{dx} \):
\[ 2by \frac{dy}{dx} = -2ax \]
Dividing through by \( 2b \):
\[ \frac{dy}{dx} = -\frac{ax}{by} \]

### Step 4: Differentiate again
Now we differentiate \( \frac{dy}{dx} \) again with respect to \( x \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(-\frac{ax}{by}\right) \]
Using the quotient rule:
\[ \frac{d^2y}{dx^2} = -\frac{b \cdot a \cdot y + a \cdot b \cdot \frac{dy}{dx} \cdot x}{b^2y^2} \]
This results in a second-order derivative.

### Step 5: Identify the order and degree
The highest derivative present in the equation after differentiation is \( \frac{d^2y}{dx^2} \), which indicates that the order of the differential equation is 2.

The degree of a differential equation is defined as the power of the highest derivative in the equation when the equation is a polynomial in derivatives. Here, the highest derivative \( \frac{d^2y}{dx^2} \) appears to the first power, so the degree is 1.

### Conclusion
Thus, the order and degree of the differential equation of the family of curves \( ax^2 + by^2 = 2 \) are:
- **Order:** 2
- **Degree:** 1