If m and n are the order and degree of the differential equation `(y'')^(5)+4 . ((y'')^(3))/(y''')+y'''=sin x, then`

Options

A. m = 3, n = 5
B. m = 3, n = 1
C. m = 3, n = 3
D. m = 3, n = 2

Explanation

To solve the given differential equation and find the order (m) and degree (n), we will follow these steps:

### Step 1: Identify the given differential equation
The differential equation is given as:
\[ (y'')^5 + 4 \cdot \frac{(y'')^3}{y'''} + y''' = \sin x \]

### Step 2: Multiply the entire equation by \(y'''\)
To eliminate the denominator, we multiply the entire equation by \(y'''\):
\[ y''' \cdot (y'')^5 + 4 \cdot (y'')^3 + (y''')^2 = \sin x \cdot y''' \]

### Step 3: Rewrite the equation
After multiplying, the equation becomes:
\[ (y'')^5 \cdot y''' + 4 \cdot (y'')^3 + (y''')^2 = \sin x \cdot y''' \]

### Step 4: Determine the order of the differential equation (m)
The order of a differential equation is defined as the highest derivative present in the equation. In our rewritten equation, the highest derivative is \(y'''\). Therefore, the order \(m\) is:
\[ m = 3 \]

### Step 5: Determine the degree of the differential equation (n)
The degree of a differential equation is defined as the power of the highest order derivative when the equation is a polynomial in derivatives. In our case, the highest order derivative is \(y'''\) and it appears with a power of 1 in the term \((y''')^2\) and also as a factor in the other terms. Therefore, the degree \(n\) is:
\[ n = 2 \]

### Final Result
Thus, we have:
\[ m = 3 \quad \text{and} \quad n = 2 \]

### Conclusion
The values of \(m\) and \(n\) are:
- Order \(m = 3\)
- Degree \(n = 2\)