The order of the diferential equation whose general solution is given by
` y = c_(1)e^(2x+c_(2)) + c_(3)e^(x)+c_(4)sin (x+c_(5))` is

Options

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

Explanation

To determine the order of the differential equation whose general solution is given by

\[ y = c_1 e^{2x + c_2} + c_3 e^{x} + c_4 \sin(x + c_5) \]

we need to analyze the expression and identify the number of arbitrary constants present.

### Step-by-Step Solution:

1. **Identify the General Solution**:
The general solution provided is:
\[ y = c_1 e^{2x + c_2} + c_3 e^{x} + c_4 \sin(x + c_5) \]

2. **Recognize the Constants**:
In the given expression, we can see the following arbitrary constants:
- \( c_1 \)
- \( c_2 \)
- \( c_3 \)
- \( c_4 \)
- \( c_5 \)

3. **Count the Arbitrary Constants**:
There are a total of 5 arbitrary constants in the solution.

4. **Determine the Order**:
The order of a differential equation is equal to the number of arbitrary constants in its general solution. Therefore, since there are 5 arbitrary constants, the order of the differential equation is:
\[ \text{Order} = 5 \]

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

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