The order of the differential equation of the family of curves `y=a3^(bx+c)+dsin(x+e)` is (where, a, b, c, d, e are arbitrary constants)

Options

A. 5
B. 4
C. 3
D. 2

Answer

To determine the order of the differential equation of the family of curves given by the equation \( y = a \cdot 3^{(bx+c)} + d \sin(x+e) \), we need to analyze the number of arbitrary constants present in the equation.

### Step-by-Step Solution:

1. **Identify the Constants**:
The equation contains the following arbitrary constants: \( a, b, c, d, e \).

2. **Count the Constants**:
We have a total of 5 arbitrary constants:
- \( a \)
- \( b \)
- \( c \)
- \( d \)
- \( e \)

3. **Understanding the Order of the Differential Equation**:
The order of a differential equation is determined by the number of arbitrary constants in the family of curves. Each arbitrary constant typically requires one differentiation to eliminate it when forming the differential equation.

4. **Form the Differential Equation**:
Since we have 5 arbitrary constants, we will need to differentiate the given equation 5 times to eliminate all constants and arrive at a differential equation.

5. **Conclusion**:
Therefore, the order of the differential equation corresponding to the given family of curves is 5.

### Final Answer:
The order of the differential equation is **5**.