What is the differential equation for `y^(2) = 4a (x - a) `?

Options

A. `yy' - 2xyy' + y^(2) = 0 `
B. `y y' (y y' + 2x) + y^(2) = 0`
C. `yy' (yy' - 2x) + y^(2) = 0`
D. `y y' - 2x y y' + y = 0`

Answer

To find the differential equation for the given equation \( y^2 = 4a(x - a) \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ y^2 = 4a(x - a) \]
We will differentiate both sides with respect to \( x \).

### Step 2: Apply implicit differentiation
Differentiating the left side:
\[ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \]
Differentiating the right side:
\[ \frac{d}{dx}(4a(x - a)) = 4a \cdot 1 = 4a \]
So, we have:
\[ 2y \frac{dy}{dx} = 4a \]

### Step 3: Solve for \( a \)
From the equation \( 2y \frac{dy}{dx} = 4a \), we can solve for \( a \):
\[ a = \frac{y}{2} \frac{dy}{dx} \]

### Step 4: Substitute \( a \) back into the original equation
Substituting \( a \) back into the original equation:
\[ y^2 = 4\left(\frac{y}{2} \frac{dy}{dx}\right)(x - \frac{y}{2} \frac{dy}{dx}) \]
This simplifies to:
\[ y^2 = 2y \frac{dy}{dx} (x - \frac{y}{2} \frac{dy}{dx}) \]

### Step 5: Expand and rearrange the equation
Expanding the right side:
\[ y^2 = 2xy \frac{dy}{dx} - y \left(\frac{dy}{dx}\right)^2 \]
Rearranging gives:
\[ y^2 + y \left(\frac{dy}{dx}\right)^2 - 2xy \frac{dy}{dx} = 0 \]

### Step 6: Factor the equation
Factoring out \( y \) and \( \frac{dy}{dx} \):
\[ y \left( \frac{dy}{dx} \right)^2 - 2xy \frac{dy}{dx} + y^2 = 0 \]
This can be factored as:
\[ y \left( \frac{dy}{dx} - 2x \right) + y^2 = 0 \]

### Final Differential Equation
Thus, the final differential equation is:
\[ y \left( \frac{dy}{dx} - 2x \right) + y^2 = 0 \]