Write the order of the differential equation associated with the primitive `y=C_1+C_2e^x+C_3e^(-2x+C_4),\ w h e r e\ C_1, C_2, C-3,\ C_4` are arbitrary constants.

Options

A. 3
B. 4
C. 2
D. none of these

Answer

To determine the order of the differential equation associated with the given primitive \( y = C_1 + C_2 e^x + C_3 e^{-2x} + C_4 \), where \( C_1, C_2, C_3, C_4 \) are arbitrary constants, we can follow these steps:

### Step 1: Identify the number of arbitrary constants
The given equation has four arbitrary constants: \( C_1, C_2, C_3, \) and \( C_4 \).

### Step 2: Understand the relationship between arbitrary constants and the order of the differential equation
The order of a differential equation is determined by the number of arbitrary constants present in its general solution. Each arbitrary constant typically corresponds to one order of differentiation.

### Step 3: Simplify the expression if necessary
In this case, we can observe that the term \( C_3 e^{-2x} \) can be combined with \( C_4 \) in a way that does not change the number of arbitrary constants. However, since we are only interested in the total number of arbitrary constants, we can keep them as they are.

### Step 4: Count the distinct arbitrary constants
After reviewing the equation, we see that we still have four distinct arbitrary constants: \( C_1, C_2, C_3, \) and \( C_4 \).

### Step 5: Conclude the order of the differential equation
Since there are four arbitrary constants, the order of the differential equation associated with this primitive is 4.

Thus, the order of the differential equation is **4**.

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