Write order of the differential equation of the family of following curves
`Ay= Bx^2`

Answer

To determine the order of the differential equation of the family of curves given by the equation \( Ay = Bx^2 \), we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ Ay = Bx^2 \]
We can rearrange this to express \( y \) in terms of \( x \):
\[ y = \frac{B}{A} x^2 \]

### Step 2: Identify constants
In the equation \( y = \frac{B}{A} x^2 \), we see that \( A \) and \( B \) are constants. We can denote \( \frac{B}{A} \) as a new constant \( k \):
\[ y = kx^2 \]
where \( k = \frac{B}{A} \).

### Step 3: Differentiate the equation
Next, we differentiate \( y \) with respect to \( x \) to form a differential equation:
\[ \frac{dy}{dx} = 2kx \]

### Step 4: Express the differential equation
We can express the relationship we found in terms of \( A \) and \( B \):
\[ \frac{dy}{dx} = \frac{2B}{A} x \]

### Step 5: Identify the order of the differential equation
The highest derivative present in the equation \( \frac{dy}{dx} = \frac{2B}{A} x \) is the first derivative \( \frac{dy}{dx} \). Therefore, the order of the differential equation is:
\[ \text{Order} = 1 \]

### Step 6: Confirm using the number of constants
We can also confirm the order of the differential equation by counting the number of constants in the original equation \( Ay = Bx^2 \). Here, we have two constants \( A \) and \( B \). However, since we can express \( y \) in terms of a single constant \( k \) (where \( k = \frac{B}{A} \)), we conclude that:
\[ \text{Order of the differential equation} = \text{Number of constants} - 1 = 2 - 1 = 1 \]

### Final Answer
Thus, the order of the differential equation of the family of curves \( Ay = Bx^2 \) is:
\[ \text{Order} = 1 \]