The differential eqaution of the family of curve `y^(2)=4a(x+a)`, is

Options

A. `y^(2)=4(dy)/(dx)(x+(dy)/(dx))`
B. `2y=(dy)/(dx)+4a`
C. `y^(2)((dy)/(dx))^(2)+2xy(dy)/(dx)-y^(2)=0`
D. `y^(2)(dy)/(dx)+4y=0`

Answer

To find the differential equation of the family of curves given by the equation \( y^2 = 4a(x + a) \), we will follow these steps:

### Step 1: Write the given equation
The equation of the family of curves is:
\[ y^2 = 4a(x + a) \]

### Step 2: Differentiate the equation with respect to \( x \)
We will differentiate both sides of the equation with respect to \( x \). Using implicit differentiation, we have:
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4a(x + a)) \]
Using the chain rule on the left side and the product rule on the right side, we get:
\[ 2y \frac{dy}{dx} = 4a \left(1 + 0\right) = 4a \]

### Step 3: Solve for \( a \)
From the differentiated equation, we can express \( a \) in terms of \( y \) and \( \frac{dy}{dx} \):
\[ a = \frac{1}{2y} \frac{dy}{dx} \]

### Step 4: Substitute \( a \) back into the original equation
Now we will substitute \( a \) back into the original equation:
\[ y^2 = 4\left(\frac{1}{2y} \frac{dy}{dx}\right)(x + \frac{1}{2y} \frac{dy}{dx}) \]
This simplifies to:
\[ y^2 = 2\frac{dy}{dx}(x + \frac{1}{2y} \frac{dy}{dx}) \]

### Step 5: Simplify the equation
Distributing the terms on the right side:
\[ y^2 = 2\frac{dy}{dx}x + \frac{1}{y} \left(\frac{dy}{dx}\right)^2 \]
Multiplying through by \( y \) to eliminate the fraction gives:
\[ y^3 = 2y\frac{dy}{dx}x + \left(\frac{dy}{dx}\right)^2 \]

### Step 6: Rearranging to form a differential equation
Rearranging the above equation leads to:
\[ \left(\frac{dy}{dx}\right)^2 - 2xy\frac{dy}{dx} + y^3 = 0 \]

### Final Differential Equation
Thus, the differential equation of the family of curves is:
\[ \left(\frac{dy}{dx}\right)^2 - 2xy\frac{dy}{dx} + y^3 = 0 \]

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