What is the degree of the differential equation `((d^(4)y)/(dx^(4)))^(3/2) -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

Answer

To find the degree of the given differential equation

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

we will follow these steps:

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

### Step 2: Determine the exponent of the highest derivative
In the equation, the term \(\left(\frac{d^4y}{dx^4}\right)^{\frac{3}{2}}\) has an exponent of \(\frac{3}{2}\).

### Step 3: Eliminate the fractional exponent
To find the degree of the differential equation, we need to eliminate the fractional exponent. We can do this by raising both sides of the equation to the power of \(2\) (since \(\frac{3}{2}\) is a fractional exponent).

This gives us:

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

### Step 4: Identify the degree
After eliminating the fractional exponent, the highest derivative \(\frac{d^4y}{dx^4}\) now has an exponent of \(3\). Therefore, the degree of the differential equation is \(3\).

### Final Answer
The degree of the differential equation is \(3\).

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