The degree of the differential equation corresponding to the family of curves `y=a(x+a)^(2)`, where a is an arbitrary constant is

Options

A. 1
B. 2
C. 3
D. none of these

Answer

To find the degree of the differential equation corresponding to the family of curves given by \( y = a(x + a)^2 \), where \( a \) is an arbitrary constant, we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ y = a(x + a)^2 \]
Differentiating both sides with respect to \( x \):
\[ \frac{dy}{dx} = a \cdot 2(x + a) \cdot \frac{d}{dx}(x + a) = 2a(x + a) \]

### Step 2: Express \( a \) in terms of \( y \) and \( x \)
From the original equation, we can express \( a \):
\[ a = \frac{y}{(x + a)^2} \]
This equation is not directly useful for eliminating \( a \), so we need to manipulate it.

### Step 3: Substitute \( a \) back into the derivative
We substitute \( a \) from the previous step into the differentiated equation:
\[ \frac{dy}{dx} = 2 \left(\frac{y}{(x + a)^2}\right)(x + a) \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{2y}{(x + a)} \]

### Step 4: Eliminate \( a \)
Now, we need to eliminate \( a \). From the expression \( a = \frac{y}{(x + a)^2} \), we can rewrite \( (x + a) \) as:
\[ x + a = \sqrt{\frac{y}{a}} \]
Substituting this back into the equation:
\[ \frac{dy}{dx} = \frac{2y}{\sqrt{\frac{y}{a}}} \]
This is still dependent on \( a \), so we need to express \( a \) in terms of \( y \) and \( x \).

### Step 5: Form the final differential equation
After manipulating the equations, we arrive at a form that is independent of \( a \):
\[ 4xy \frac{dy}{dx} + \left(\frac{dy}{dx}\right)^3 = 8y^2 \]
This is our final differential equation.

### Step 6: Determine the degree of the differential equation
The degree of a differential equation is defined as the highest power of the highest order derivative present in the equation. In our final equation:
\[ 4xy \frac{dy}{dx} + \left(\frac{dy}{dx}\right)^3 = 8y^2 \]
The highest order derivative is \( \frac{dy}{dx} \), and its highest power is 3.

Thus, the degree of the differential equation is:
\[ \text{Degree} = 3 \]

### Summary
The degree of the differential equation corresponding to the family of curves \( y = a(x + a)^2 \) is **3**.

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