The order of the differential equation of the family of curves `y=(a)/(c ) sin (bx)+3^(dx)` where a, b, c, d are arbitrary constants is

Options

A. 4
B. 3
C. 2
D. 1

Answer

To find the order of the differential equation of the family of curves given by the equation

\[ y = \frac{a}{c} \sin(bx) + 3^{dx} \]

where \( a, b, c, d \) are arbitrary constants, we will follow these steps:

### Step 1: Identify the constants
In the given equation, we have the following constants:
- \( \frac{a}{c} \) (which can be treated as one constant)
- \( b \)
- \( d \)

### Step 2: Count the distinct constants
From the equation, we can see that:
- \( \frac{a}{c} \) can be treated as a single constant.
- \( b \) is another distinct constant.
- \( d \) is also a distinct constant.

Thus, we have a total of **3 distinct constants**.

### Step 3: Determine the order of the differential equation
The order of a differential equation is determined by the number of arbitrary constants present in the family of curves. Since we have identified **3 distinct constants**, the order of the differential equation will be **3**.

### Conclusion
Therefore, the order of the differential equation of the family of curves is **3**.