The order and degree of the differential equation `((d^(2)y)/(dx^2))^(3"/"2)- sqrt(((dy)/(dx)))-4=0` are respectively

Options

A. 2 and 6
B. 3 and 6
C. 1 and 4
D. 2 and 4

Answer

To determine the order and degree of the given differential equation:

\[ \left(\frac{d^2y}{dx^2}\right)^{\frac{3}{2}} - \sqrt{\frac{dy}{dx}} - 4 = 0 \]

we will follow these steps:

### Step 1: Identify the highest derivative
The highest derivative in the equation is \(\frac{d^2y}{dx^2}\), which is the second derivative of \(y\).

### Step 2: Determine the order
The order of a differential equation is defined as the highest order of the derivative present in the equation. Since the highest derivative here is \(\frac{d^2y}{dx^2}\), which is of order 2, the order of the differential equation is:

\[ \text{Order} = 2 \]

### Step 3: Rewrite the equation for degree calculation
Next, we need to express the equation in a form where we can identify the degree. The degree of a differential equation is defined as the power of the highest order derivative when the equation is a polynomial in derivatives.

To simplify, we can rewrite the equation as:

\[ \left(\frac{d^2y}{dx^2}\right)^{\frac{3}{2}} = \sqrt{\frac{dy}{dx}} + 4 \]

### Step 4: Eliminate the roots by squaring
To eliminate the fractional exponent and the square root, we can square both sides of the equation:

\[ \left(\left(\frac{d^2y}{dx^2}\right)^{\frac{3}{2}}\right)^2 = \left(\sqrt{\frac{dy}{dx}} + 4\right)^2 \]

This leads to:

\[ \left(\frac{d^2y}{dx^2}\right)^3 = \left(\frac{dy}{dx} + 8\right) + 16 \]

### Step 5: Rearranging the equation
After squaring, we can rearrange the equation:

\[ \left(\frac{d^2y}{dx^2}\right)^3 - \frac{dy}{dx} - 16 = 0 \]

### Step 6: Identify the degree
Now, we can see that the highest derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 3. Therefore, the degree of the differential equation is:

\[ \text{Degree} = 3 \]

### Conclusion
Thus, the order and degree of the differential equation are:

\[ \text{Order} = 2, \quad \text{Degree} = 3 \]