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 can follow these steps:

### Step 1: Rewrite the given equation
Start with the equation of the family of curves:
\[ y = a(x + a)^2 \]

### Step 2: Differentiate with respect to \( x \)
Differentiate both sides with respect to \( x \) to eliminate the constant \( a \):
\[ \frac{dy}{dx} = \frac{d}{dx}[a(x + a)^2] \]
Using the product rule and chain rule:
\[ \frac{dy}{dx} = a \cdot 2(x + a) \cdot \frac{d}{dx}(x + a) = 2a(x + a) \]

### Step 3: Solve for \( a \)
From the equation \( \frac{dy}{dx} = 2a(x + a) \), we can express \( a \) in terms of \( y \) and \( x \):
\[ a = \frac{1}{2(x + a)}\frac{dy}{dx} \]
To eliminate \( a \), we can substitute \( a \) back into the original equation.

### Step 4: Substitute \( a \) back into the original equation
Substituting \( a \) into the original equation:
\[ y = \frac{1}{2(x + a)}\frac{dy}{dx}(x + a)^2 \]
This substitution will lead to a more complex expression, but we will focus on the degree of the resulting differential equation.

### Step 5: Differentiate again
Differentiate \( \frac{dy}{dx} = 2a(x + a) \) again to find the second derivative:
\[ \frac{d^2y}{dx^2} = 2\left(\frac{da}{dx}(x + a) + a\right) \]
This equation will involve \( a \) and its derivatives.

### Step 6: Identify the highest order derivative
The highest order derivative in the final differential equation will be \( \frac{d^2y}{dx^2} \). The degree of a differential equation is defined as the power of the highest order derivative when the equation is a polynomial in derivatives.

### Conclusion
The highest order derivative is \( \frac{d^2y}{dx^2} \), and it appears to the first power. Therefore, the degree of the differential equation corresponding to the family of curves \( y = a(x + a)^2 \) is:

\[ \text{Degree} = 1 \]