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Step-by-Step Solution
Step 1: Write down the given family of curves
The family of curves is given by
y^2 = a\Bigl(x + \frac{\sqrt{a}}{2}\Bigr), \quad a > 0.
Here, a is the parameter that generates different curves in the family.
Step 2: Differentiate both sides with respect to x
Differentiate implicitly:
\frac{d}{dx}\bigl(y^2\bigr) \;=\; \frac{d}{dx}\Bigl[a\Bigl(x + \frac{\sqrt{a}}{2}\Bigr)\Bigr].
This gives:
2y\,\frac{dy}{dx} \;=\; a.
From here,
a \;=\; 2y\,\frac{dy}{dx}.
Step 3: Eliminate the parameter a
Substitute a = 2y\,\frac{dy}{dx} back into the original equation y^2 = a\bigl(x + \tfrac{\sqrt{a}}{2}\bigr) . This yields:
y^2 \;=\; 2y\,\frac{dy}{dx}\biggl(x \;+\; \frac{\sqrt{2y\,\frac{dy}{dx}}}{2}\biggr).
Note that \sqrt{a} = \sqrt{2y\,\frac{dy}{dx}} = \sqrt{2}\,\sqrt{y\,\frac{dy}{dx}}.
Step 4: Rearrange and isolate terms
Distributing and rearranging step by step leads (after appropriate algebraic manipulations) to the form:
y - 2x\,\frac{dy}{dx} \;=\; \sqrt{2}\,\frac{dy}{dx}\,\sqrt{y\,\frac{dy}{dx}}.
Squaring both sides:
(\,y \;-\; 2x\,\frac{dy}{dx}\,)^{2} \;=\; 2\,y\,\bigl(\tfrac{dy}{dx}\bigr)^{3}.
Step 5: Identify the order and degree of the differential equation
Order: The highest order of derivative in the final equation is 1 (since only \frac{dy}{dx} appears).
Degree: The highest power to which \frac{dy}{dx} is raised in the final equation is 3.
Thus,
\text{Order} = 1, \quad \text{Degree} = 3.
Step 6: Compute the difference between degree and order
The required difference is
\text{Degree} - \text{Order} = 3 - 1 = 2.
Therefore, the difference between the degree and order of the differential equation is 2.
