What is the degree of the differential equation.
` ((d^(4)y)/(dx^(4)))^(3//5) -5 (d^(3)y)/(dx^(3)) +6(d^(2)y)/(dx^(2))-8 (dy)/(dx)+5=0` ?

Options

A. 5
B. 4
C. 3
D. 2

Explanation

To determine the degree of the given differential equation:

\[ \left(\frac{d^4y}{dx^4}\right)^{\frac{3}{5}} - 5 \frac{d^3y}{dx^3} + 6 \frac{d^2y}{dx^2} - 8 \frac{dy}{dx} + 5 = 0 \]

we will follow these steps:

### Step 1: Identify the highest order derivative
The highest order derivative in the equation is \(\frac{d^4y}{dx^4}\).

### Step 2: Rewrite the equation
To analyze the degree, we can rewrite the equation as:

\[ \left(\frac{d^4y}{dx^4}\right)^{\frac{3}{5}} = 5 \frac{d^3y}{dx^3} - 6 \frac{d^2y}{dx^2} + 8 \frac{dy}{dx} - 5 \]

### Step 3: Eliminate fractional powers
To eliminate the fractional power, we raise both sides of the equation to the power of 5:

\[ \left(\frac{d^4y}{dx^4}\right)^3 = \left(5 \frac{d^3y}{dx^3} - 6 \frac{d^2y}{dx^2} + 8 \frac{dy}{dx} - 5\right)^5 \]

### Step 4: Identify the degree
The degree of a differential equation is defined as the power of the highest order derivative after the equation has been simplified to remove any fractional or negative powers.

In our case, after raising to the power of 5, the highest order derivative \(\frac{d^4y}{dx^4}\) is raised to the power of 3. Therefore, the degree of the differential equation is:

\[ \text{Degree} = 3 \]

### Final Answer
Thus, the degree of the given differential equation is **3**.

---