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 find the degree of the differential equation \( Y_{2}^{\frac{3}{2}} - Y_{1}^{\frac{1}{2}} - 4 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation in terms of derivatives
The notation \( Y_{2} \) and \( Y_{1} \) represents derivatives of \( y \):
- \( Y_{2} = \frac{d^2y}{dx^2} \)
- \( Y_{1} = \frac{dy}{dx} \)

Thus, we can rewrite the equation as:
\[ \left(\frac{d^2y}{dx^2}\right)^{\frac{3}{2}} - \left(\frac{dy}{dx}\right)^{\frac{1}{2}} - 4 = 0 \]

### Step 2: Isolate the highest order derivative
Rearranging the equation gives:
\[ \left(\frac{d^2y}{dx^2}\right)^{\frac{3}{2}} = \left(\frac{dy}{dx}\right)^{\frac{1}{2}} + 4 \]

### Step 3: Eliminate fractional powers
To eliminate the fractional powers, we will square both sides of the equation:
\[ \left(\left(\frac{d^2y}{dx^2}\right)^{\frac{3}{2}}\right)^{2} = \left(\left(\frac{dy}{dx}\right)^{\frac{1}{2}} + 4\right)^{2} \]
This simplifies to:
\[ \left(\frac{d^2y}{dx^2}\right)^{3} = \left(\frac{dy}{dx}\right) + 8\left(\frac{dy}{dx}\right)^{\frac{1}{2}} + 16 \]

### Step 4: Isolate the highest order derivative again
Now we can rearrange the equation:
\[ \left(\frac{d^2y}{dx^2}\right)^{3} - \left(\frac{dy}{dx}\right) - 16 = 8\left(\frac{dy}{dx}\right)^{\frac{1}{2}} \]

### Step 5: Square both sides again to eliminate the square root
Squaring both sides again:
\[ \left(\left(\frac{d^2y}{dx^2}\right)^{3} - \left(\frac{dy}{dx}\right) - 16\right)^{2} = (8\left(\frac{dy}{dx}\right)^{\frac{1}{2}})^{2} \]
This gives:
\[ \left(\frac{d^2y}{dx^2}\right)^{6} - 2\left(\frac{d^2y}{dx^2}\right)^{3}\left(\frac{dy}{dx}\right) - 32\left(\frac{d^2y}{dx^2}\right)^{3} + \left(\frac{dy}{dx}\right)^{2} + 256 = 64\left(\frac{dy}{dx}\right) \]

### 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. Here, the highest order derivative is \( \frac{d^2y}{dx^2} \) and its highest power is \( 6 \).

Thus, the degree of the differential equation is:
\[ \text{Degree} = 6 \]

### Final Answer
The degree of the differential equation \( Y_{2}^{\frac{3}{2}} - Y_{1}^{\frac{1}{2}} - 4 = 0 \) is **6**.

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