Form the differential equation for the family of curves` y^2= 4a (x+b)`where a and b arbitrary constants.

Answer

To form the differential equation for the family of curves given by the equation \( y^2 = 4a(x + b) \), where \( a \) and \( b \) are arbitrary constants, we will follow these steps:

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

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

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

Using the chain rule on the left side and the product rule on the right side, we get:
\[ 2y \frac{dy}{dx} = 4a \cdot 1 \]

This simplifies to:
\[ 2y \frac{dy}{dx} = 4a \]

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

From the equation \( 2y \frac{dy}{dx} = 4a \), we can express \( a \) in terms of \( y \) and \( \frac{dy}{dx} \):
\[ a = \frac{y \frac{dy}{dx}}{2} \]

### Step 3: Differentiate again to eliminate \( a \)

Next, we differentiate the equation \( y^2 = 4a(x + b) \) again with respect to \( x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4a(x + b)) \]

Using the product rule on the right side:
\[ 2y \frac{dy}{dx} = 4 \left( \frac{da}{dx}(x + b) + a \cdot 1 \right) \]

### Step 4: Substitute \( a \) into the second derivative

We know \( a = \frac{y \frac{dy}{dx}}{2} \), so substituting this into the equation gives:
\[ 2y \frac{dy}{dx} = 4 \left( \frac{da}{dx}(x + b) + \frac{y \frac{dy}{dx}}{2} \right) \]

### Step 5: Differentiate \( a \)

To find \( \frac{da}{dx} \), we differentiate \( a = \frac{y \frac{dy}{dx}}{2} \):
\[ \frac{da}{dx} = \frac{1}{2} \left( \frac{dy}{dx} \frac{dy}{dx} + y \frac{d^2y}{dx^2} \right) = \frac{1}{2} \left( \left(\frac{dy}{dx}\right)^2 + y \frac{d^2y}{dx^2} \right) \]

### Step 6: Substitute \( \frac{da}{dx} \) back into the equation

Substituting this back into our equation, we can simplify and eliminate \( a \).

### Final Differential Equation

After simplification, we arrive at the final form of the differential equation:
\[ y \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 0 \]

This is the required differential equation for the family of curves.

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