Key Concepts and Formulas

• Order of a Differential Equation: The order of the highest derivative appearing in the differential equation.
• Degree of a Differential Equation: The power of the highest order derivative in the differential equation, after the equation has been made free of radicals and fractions in the derivatives.
• Formation of Differential Equation: Eliminate arbitrary constants by differentiation and substitution.

Step-by-Step Solution

Step 1: Identify the Arbitrary Constant and Differentiate

The given equation is: $y^2 = a\left(x + \frac{\sqrt{a}}{2}\right)$ We identify $a$ as the arbitrary constant. Since there is one arbitrary constant, we need to differentiate the equation once with respect to $x$ to eliminate $a$.

Differentiating 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 on the left side and treating $a$ as a constant on the right side: $2y \frac{dy}{dx} = a \frac{d}{dx}\left(x + \frac{\sqrt{a}}{2}\right)$ Since $\frac{\sqrt{a}}{2}$ is a constant, its derivative with respect to $x$ is 0: $2y \frac{dy}{dx} = a(1 + 0)$ $2y \frac{dy}{dx} = a$ Let $y' = \frac{dy}{dx}$. Then, $a = 2yy'$

Step 2: Eliminate the Arbitrary Constant

Substitute $a = 2yy'$ back into the original equation: $y^2 = (2yy')\left(x + \frac{\sqrt{2yy'}}{2}\right)$

Step 3: Simplify and Remove Radicals

Simplify the equation: $y^2 = 2xyy' + 2yy'\frac{\sqrt{2yy'}}{2}$ $y^2 = 2xyy' + yy'\sqrt{2yy'}$ Assuming $y \neq 0$, divide by $y$: $y = 2xy' + y'\sqrt{2yy'}$ Isolate the radical term: $y - 2xy' = y'\sqrt{2yy'}$ Square both sides: $(y - 2xy')^2 = (y')^2 (2yy')$ $y^2 - 4xyy' + 4x^2(y')^2 = 2y(y')^3$ $y^2 - 4xy\frac{dy}{dx} + 4x^2\left(\frac{dy}{dx}\right)^2 = 2y\left(\frac{dy}{dx}\right)^3$

Step 4: Determine the Order and Degree

The highest order derivative in the differential equation is $\frac{dy}{dx}$, which is the first derivative. Therefore, the order of the differential equation is 1.

The power of the highest order derivative, $\frac{dy}{dx}$, is 3 (in the term $2y(\frac{dy}{dx})^3$). Therefore, the degree of the differential equation is 3.

Step 5: Calculate the Difference

The difference between the degree and the order is: $\text{Degree} - \text{Order} = 3 - 1 = 2$

Common Mistakes & Tips

• Remember to eliminate the arbitrary constant by substituting its value back into the original equation.
• Make sure the equation is free of radicals and fractions involving derivatives before determining the degree.
• Carefully apply the chain rule when differentiating.

Summary

We started with the given equation $y^2 = a\left(x + \frac{\sqrt{a}}{2}\right)$ and differentiated it once with respect to $x$ to get an expression for $a$ in terms of $y$ and $y'$. We substituted this expression back into the original equation to eliminate $a$. After simplifying and removing the radical, we obtained a differential equation. Finally, we identified the order as 1 and the degree as 3, and calculated their difference, which is 2.

Final Answer

The final answer is \boxed{2}.