The order and degree of the differential equation `(d^4 y)/(dx^(4))+ sin(y''')=0` are respectively

Options

A. 4 and 1
B. 1 and 2
C. 4 and 4
D. 4 and not defined

Answer

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

\[ \frac{d^4 y}{dx^4} + \sin(y''') = 0 \]

### Step 1: Identify the highest derivative
The first step is to identify the highest derivative present in the equation.

- In this equation, we have:
- The term \(\frac{d^4 y}{dx^4}\) represents the fourth derivative of \(y\).
- The term \(y'''\) represents the third derivative of \(y\).

Since \(\frac{d^4 y}{dx^4}\) is the highest derivative, the order of the differential equation is **4**.

### Step 2: Determine the degree of the differential equation
Next, we need to determine the degree of the differential equation. The degree is defined as the power of the highest order derivative when the equation is a polynomial in derivatives.

- In this case, the equation includes \(\sin(y''')\), which is a non-polynomial function of the third derivative \(y'''\). Since the presence of the sine function makes it non-polynomial, we cannot define a degree for this differential equation.

Therefore, the degree of the differential equation is **not defined**.

### Final Answer
- **Order**: 4
- **Degree**: Not defined