`y=c_1 x+c_2 sin(2x+c_3)` (`C_1, C_2, C_3` are arbitrary constants)

Options

A. Order of differential equation is 2
B. Order of differential equation is 3
C. Degree of differential equation is 1
D. The differential equation is `(yd^(3)y)/(dx^(3)) = (dy)/(dx).(d^(2)y)/(dx^(2))`

Answer

To solve the problem, we need to analyze the given function and derive the necessary differential equations to determine the order and degree of the differential equation.

### Step-by-Step Solution:

1. **Given Function**:
\[ y = c_1 x + c_2 \sin(2x + c_3) \]
where \(c_1\), \(c_2\), and \(c_3\) are arbitrary constants.

2. **First Derivative**:
Differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = c_1 + c_2 \cdot \frac{d}{dx}(\sin(2x + c_3)) \]
Using the chain rule:
\[ \frac{dy}{dx} = c_1 + c_2 \cdot \cos(2x + c_3) \cdot 2 = c_1 + 2c_2 \cos(2x + c_3) \]

3. **Second Derivative**:
Differentiate \(\frac{dy}{dx}\) with respect to \(x\):
\[ \frac{d^2y}{dx^2} = 0 + 2c_2 \cdot \frac{d}{dx}(\cos(2x + c_3)) \]
Again using the chain rule:
\[ \frac{d^2y}{dx^2} = 2c_2 \cdot (-\sin(2x + c_3)) \cdot 2 = -4c_2 \sin(2x + c_3) \]

4. **Third Derivative**:
Differentiate \(\frac{d^2y}{dx^2}\) with respect to \(x\):
\[ \frac{d^3y}{dx^3} = -4c_2 \cdot \frac{d}{dx}(\sin(2x + c_3)) \]
Using the chain rule:
\[ \frac{d^3y}{dx^3} = -4c_2 \cdot \cos(2x + c_3) \cdot 2 = -8c_2 \cos(2x + c_3) \]

5. **Forming the Differential Equation**:
Now we have:
- \(y = c_1 x + c_2 \sin(2x + c_3)\)
- \(\frac{dy}{dx} = c_1 + 2c_2 \cos(2x + c_3)\)
- \(\frac{d^2y}{dx^2} = -4c_2 \sin(2x + c_3)\)
- \(\frac{d^3y}{dx^3} = -8c_2 \cos(2x + c_3)\)

6. **Identifying the Order and Degree**:
The highest order of the derivative we have is the third derivative, which means the order of the differential equation is 3. The degree is determined by the highest power of the derivatives in the equation, which is 1 (since they are linear in terms of \(c_1\), \(c_2\), and \(c_3\)).

### Summary of Findings:
- **Order of the Differential Equation**: 3
- **Degree of the Differential Equation**: 1