The order of the differential equation whose general solution is `y = c_(1) cos 2x + c_(2) cos^(2) x + c_(3) sin^(2) x + c_(4)` is

Options

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

Explanation

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

\[ y = c_1 \cos(2x) + c_2 \cos^2(x) + c_3 \sin^2(x) + c_4, \]

we need to analyze the components of the solution.

### Step 1: Identify the functions involved

The general solution consists of:
- \( c_1 \cos(2x) \)
- \( c_2 \cos^2(x) \)
- \( c_3 \sin^2(x) \)
- \( c_4 \)

### Step 2: Rewrite \( \cos^2(x) \) and \( \sin^2(x) \)

Using the double angle identities, we can rewrite \( \cos^2(x) \) and \( \sin^2(x) \):

\[ \cos^2(x) = \frac{1 + \cos(2x)}{2} \]
\[ \sin^2(x) = \frac{1 - \cos(2x)}{2} \]

### Step 3: Substitute back into the equation

Substituting these identities into the general solution gives:

\[ y = c_1 \cos(2x) + c_2 \left(\frac{1 + \cos(2x)}{2}\right) + c_3 \left(\frac{1 - \cos(2x)}{2}\right) + c_4 \]

### Step 4: Simplify the expression

Now, we can simplify the expression:

\[ y = c_1 \cos(2x) + \frac{c_2}{2} + \frac{c_2}{2} \cos(2x) + \frac{c_3}{2} - \frac{c_3}{2} \cos(2x) + c_4 \]

Combining like terms:

\[ y = \left(c_1 + \frac{c_2}{2} - \frac{c_3}{2}\right) \cos(2x) + \left(c_4 + \frac{c_2}{2} + \frac{c_3}{2}\right) \]

### Step 5: Define constants

Let:
- \( a = c_4 + \frac{c_2}{2} + \frac{c_3}{2} \)
- \( b = c_1 + \frac{c_2}{2} - \frac{c_3}{2} \)

Thus, we can express \( y \) as:

\[ y = a + b \cos(2x) \]

### Step 6: Determine the order of the differential equation

The expression \( y = a + b \cos(2x) \) indicates that the solution depends on the function \( \cos(2x) \). The highest derivative that can be taken while still involving the constants \( c_1, c_2, c_3, c_4 \) is determined by the highest frequency term, which is \( \cos(2x) \).

Since \( \cos(2x) \) is a periodic function with a frequency of 2, the order of the differential equation is 2.

### Final Answer

The order of the differential equation is **2**.

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