The degree of the differential equation `Y_(2) ^(3//2)- Y_(1)^(1//2)-4=0` is :

Options

A. 6
B. 3
C. 2
D. 4

Answer

To determine the degree of the differential equation \( Y_2^{3/2} - Y_1^{1/2} - 4 = 0 \), we will follow these steps:

### Step 1: Identify the terms
In the given equation:
- \( Y_2 \) represents the second derivative of \( y \) with respect to \( x \), which is \( \frac{d^2y}{dx^2} \).
- \( Y_1 \) represents the first derivative of \( y \) with respect to \( x \), which is \( \frac{dy}{dx} \).

### Step 2: Rewrite the equation
We can rewrite the equation as:
\[ Y_2^{3/2} - Y_1^{1/2} = 4 \]
This shows the relationship between the derivatives.

### Step 3: Isolate one of the derivatives
We can isolate \( Y_1^{1/2} \):
\[ Y_2^{3/2} - 4 = Y_1^{1/2} \]

### Step 4: Square both sides
To eliminate the square roots, we square both sides:
\[ (Y_2^{3/2} - 4)^2 = Y_1 \]
Expanding the left-hand side:
\[ Y_2^3 - 8Y_2^{3/2} + 16 = Y_1 \]

### Step 5: Identify the highest power of derivatives
In this equation, we can see that:
- The highest power of \( Y_2 \) (which is \( Y_2 = \frac{d^2y}{dx^2} \)) is \( 3 \).
- The highest power of \( Y_1 \) (which is \( Y_1 = \frac{dy}{dx} \)) is \( 1 \).

### Step 6: Determine the degree
The degree of a differential equation is defined as the highest power of the highest order derivative when the equation is a polynomial in derivatives. Here, the highest order derivative is \( Y_2 \) and its highest power is \( 3 \).

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

### Final Answer
The degree of the differential equation \( Y_2^{3/2} - Y_1^{1/2} - 4 = 0 \) is **3**.