The difference between degree and order of a differential equation that represents the family of curves given by ` y^2 = a( x+(sqrt(a))/(2)) ,a gt 0` is

Answer

To find the difference between the degree and order of the differential equation representing the family of curves given by \( y^2 = a \left( x + \frac{\sqrt{a}}{2} \right) \), where \( a > 0 \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ y^2 = a \left( x + \frac{\sqrt{a}}{2} \right) \]
To express this in terms of a differential equation, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}\left(a \left( x + \frac{\sqrt{a}}{2} \right)\right) \]
Using the chain rule, we have:
\[ 2y \frac{dy}{dx} = a \]
Since \( a \) is a constant, the derivative of the right-hand side is simply:
\[ 2y \frac{dy}{dx} = a \]

### Step 2: Express \( a \) in terms of \( y \) and \( \frac{dy}{dx} \)
From the equation \( 2y \frac{dy}{dx} = a \), we can express \( a \) as:
\[ a = 2y \frac{dy}{dx} \]

### Step 3: Substitute \( a \) back into the original equation
Now, substituting \( a \) back into the original equation:
\[ y^2 = (2y \frac{dy}{dx}) \left( x + \frac{\sqrt{2y \frac{dy}{dx}}}{2} \right) \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ y^2 = 2y \frac{dy}{dx} x + \frac{(2y \frac{dy}{dx}) \sqrt{2y \frac{dy}{dx}}}{2} \]

### Step 5: Square both sides to eliminate the square root
To eliminate the square root and express the equation in a standard form, we square both sides:
\[ (y^2)^2 = \left(2y \frac{dy}{dx} x + \frac{(2y \frac{dy}{dx}) \sqrt{2y \frac{dy}{dx}}}{2}\right)^2 \]

### Step 6: Identify the order and degree of the resulting differential equation
After simplifying the squared equation, we can determine the order and degree:
- The **order** of the differential equation is the highest derivative present. In this case, we have \( \frac{dy}{dx} \) appearing, so the order is 1.
- The **degree** is determined by the highest power of the highest derivative when the equation is a polynomial in derivatives. In this case, the highest power of \( \frac{dy}{dx} \) is 2, so the degree is 2.

### Step 7: Calculate the difference between degree and order
Now, we can find the difference between the degree and order:
\[ \text{Difference} = \text{Degree} - \text{Order} = 2 - 1 = 1 \]

### Final Answer
The difference between the degree and order of the differential equation is:
\[ \boxed{1} \]