Statement 1 : The order of the differential equation whose general solution is `y==c_1cos2x+c_2sin^2x+c_3cos^2x+c_4e^(2x)+c_5e^(2x+c_6)` is 3. Statement 2 : Total number of arbitrary parameters in the given general solution in the statement (1) is 3.

Options

A. Statement I is true ,and Statement II is the correct explanation for Statement I.
B. Statement I is true, Statement II is true and Statement II is the correct explanation for Statment I
C. Statement I is true, Statement II is false.
D. Statement I is false, Statement II is true.

Answer

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

\[ y = c_1 \cos(2x) + c_2 \sin^2(x) + c_3 \cos^2(x) + c_4 e^{2x} + c_5 e^{2x} + c_6 \]

we will analyze the expression step by step.

### Step 1: Identify the arbitrary constants
The general solution contains the following arbitrary constants:
- \( c_1 \)
- \( c_2 \)
- \( c_3 \)
- \( c_4 \)
- \( c_5 \)
- \( c_6 \)

### Step 2: Count the arbitrary constants
From the expression, we can see that there are a total of 6 arbitrary constants.

### Step 3: 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. Therefore, since we have 6 arbitrary constants, the order of the differential equation is 6.

### Step 4: Analyze the statements
- **Statement 1** claims that the order of the differential equation is 3. This is **false** because we found that the order is 6.
- **Statement 2** claims that the total number of arbitrary parameters is 3. This is also **false** because we found that there are 6 arbitrary parameters.

### Conclusion
Both statements are incorrect.