Differential equation of `x^2= 4b(y + b)`, where b is a parameter, is

Options

A. `x((dy)/(dx))^2=2y((dy)/(dx))+x^2`
B. `x((dy)/(dx))^2=2y((dy)/(dx))+x`
C. `x((dy)/(dx))^2=y((dy)/(dx))+x^2`
D. `x((dy)/(dx))^2=2y((dy)/(dx))+2x^2`

Explanation

To find the differential equation of the given equation \( x^2 = 4b(y + b) \), where \( b \) is a parameter, we can follow these steps:

### Step 1: Differentiate the given equation with respect to \( x \)

We start with the equation:
\[ x^2 = 4b(y + b) \]

Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}(x^2) = \frac{d}{dx}(4b(y + b)) \]

Using the chain rule on the right side, we have:
\[ 2x = 4b\left(\frac{dy}{dx}\right) \]

### Step 2: Solve for \( b \)

From the differentiated equation, we can express \( b \):
\[ b = \frac{2x}{4\frac{dy}{dx}} = \frac{x}{2\frac{dy}{dx}} \]

### Step 3: Substitute \( b \) back into the original equation

Now we substitute \( b \) back into the original equation:
\[ x^2 = 4\left(\frac{x}{2\frac{dy}{dx}}\right)(y + \frac{x}{2\frac{dy}{dx}}) \]

### Step 4: Simplify the equation

This simplifies to:
\[ x^2 = 2x\left(y + \frac{x}{2\frac{dy}{dx}}\right) \]

Distributing the right side:
\[ x^2 = 2xy + \frac{x^2}{\frac{dy}{dx}} \]

### Step 5: Rearranging the equation

Rearranging gives:
\[ x^2 - 2xy = \frac{x^2}{\frac{dy}{dx}} \]

### Step 6: Multiply through by \( \frac{dy}{dx} \)

Multiplying through by \( \frac{dy}{dx} \) gives:
\[ x^2 \frac{dy}{dx} - 2xy \frac{dy}{dx} = x^2 \]

### Step 7: Rearranging to form the differential equation

Rearranging this gives us:
\[ x^2 \frac{dy}{dx} = 2xy \frac{dy}{dx} + x^2 \]

### Step 8: Final form of the differential equation

Thus, we can express the differential equation as:
\[ x \left(\frac{dy}{dx}\right)^2 = 2y \frac{dy}{dx} + x \]

This can be rewritten as:
\[ x \left(\frac{dy}{dx}\right)^2 - 2y \frac{dy}{dx} - x = 0 \]

### Final Answer:
The differential equation is:
\[ x \left(\frac{dy}{dx}\right)^2 = 2y \frac{dy}{dx} + x \]

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