The order of the differential equation of the family of curves `y=k_(1)2^(k_(2)x)+k_(3)3^(x+k_(4))` is (where, `k_(1),k_(2), k_(3), k_(4)` are arbitrary constants)

Options

A. 4
B. 5
C. 3
D. 6

Answer

To determine the order of the differential equation corresponding to the family of curves given by the equation:

\[ y = k_1 \cdot 2^{k_2 x} + k_3 \cdot 3^{x + k_4} \]

where \( k_1, k_2, k_3, k_4 \) are arbitrary constants, we can follow these steps:

### Step 1: Identify the number of arbitrary constants
The first step is to identify how many arbitrary constants are present in the given equation. The constants in the equation are \( k_1, k_2, k_3, \) and \( k_4 \).

### Step 2: Count the distinct constants
We have four constants: \( k_1, k_2, k_3, \) and \( k_4 \). However, we need to check if any of these can be combined or eliminated.

### Step 3: Analyze the equation
The term \( 3^{x + k_4} \) can be rewritten as \( 3^x \cdot 3^{k_4} \). Here, \( 3^{k_4} \) is a constant, which we can denote as \( k_4' \). Thus, we can rewrite the equation as:

\[ y = k_1 \cdot 2^{k_2 x} + k_3 \cdot k_4' \cdot 3^x \]

### Step 4: Combine constants
Now, we can define a new constant \( k' = k_3 \cdot k_4' \). This means we can reduce the number of constants from four to three:

\[ y = k_1 \cdot 2^{k_2 x} + k' \cdot 3^x \]

### Step 5: Determine the order of the differential equation
The order of the differential equation corresponds to the number of arbitrary constants that remain after simplification. Since we have reduced the number of constants to three (\( k_1, k_2, k' \)), the order of the differential equation is 3.

### Conclusion
Thus, the order of the differential equation of the family of curves is:

**Order = 3**

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