The order of the differential equation whose general solution is given by `y= (A +B) cos (x +C)+ De^(x)` is

Options

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

Answer

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

\[ y = (A + B) \cos(x + C) + D e^x, \]

we need to analyze the number of arbitrary constants present in the solution.

### Step-by-Step Solution:

1. **Identify the General Solution**:
The given general solution is:
\[ y = (A + B) \cos(x + C) + D e^x. \]

2. **Count the Arbitrary Constants**:
- In the expression \( (A + B) \), we have two constants \( A \) and \( B \). However, since \( (A + B) \) is a combination of these two constants, we can treat it as a single new constant. Let's denote this new constant as \( E = A + B \).
- The term \( C \) is another arbitrary constant.
- The term \( D \) is also an arbitrary constant.

3. **Total Number of Arbitrary Constants**:
- After combining \( A \) and \( B \) into \( E \), we have:
- \( E \) (from \( A + B \)),
- \( C \),
- \( D \).
- Therefore, the total number of arbitrary constants is 3.

4. **Determine the Order of the Differential Equation**:
- The order of a differential equation is equal to the number of arbitrary constants in its general solution.
- Since we have identified 3 arbitrary constants, the order of the differential equation is:
\[ \text{Order} = 3. \]

### Final Answer:
The order of the differential equation whose general solution is given by \( y = (A + B) \cos(x + C) + D e^x \) is **3**.