Write the order and degree of the following differential equations.
`((d^3y)/(dx^3))^2+((d^2y)/(dx^2))^3= sin x`

Options

A. order=3, degree =3
B. order=2, degree =2
C. order=3, degree =2
D. order=2, degree =3

Answer

To determine the order and degree of the given differential equation

\[ \left(\frac{d^3y}{dx^3}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^3 = \sin x, \]

we will follow these steps:

### Step 1: Identify the derivatives present in the equation.
In the given equation, we have two derivatives:
1. \(\frac{d^3y}{dx^3}\) (the third derivative of \(y\))
2. \(\frac{d^2y}{dx^2}\) (the second derivative of \(y\))

### Step 2: Determine the order of the differential equation.
The order of a differential equation is defined as the highest order of derivative present in the equation. Here, the highest order derivative is \(\frac{d^3y}{dx^3}\).

Thus, the order of the differential equation is:
\[ \text{Order} = 3 \]

### Step 3: Determine the degree of the differential equation.
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 this case, the highest order derivative is \(\frac{d^3y}{dx^3}\), and it appears squared in the equation, i.e., \(\left(\frac{d^3y}{dx^3}\right)^2\).

Thus, the degree of the differential equation is:
\[ \text{Degree} = 2 \]

### Final Answer:
- **Order**: 3
- **Degree**: 2