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

Options

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

Answer

To find the degree of the given differential equation

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

we will follow these steps:

### Step 1: Identify the highest order derivative
The highest order derivative in the equation is \(\frac{d^3 y}{dx^3}\).

### Step 2: Rewrite the equation
To analyze the degree, we can isolate the highest order derivative. We can rearrange the equation as follows:

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

### Step 3: Eliminate the fractional exponent
To eliminate the fractional exponent, we can raise both sides of the equation to the power of \(3\):

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

### Step 4: Determine the degree
The degree of a differential equation is defined as the power of the highest order derivative after the equation has been made polynomial in derivatives. In this case, the highest order derivative \(\frac{d^3 y}{dx^3}\) is raised to the power of \(2\). Therefore, the degree of the differential equation is \(2\).

### Final Answer
Thus, the degree of the given differential equation is \(2\).

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