The order of the differential equation whose general solution is `y = (C_(1) + C_(2)) cos (x + C_(3)) - C_(4)e^(x^(4))` where `C_(1), C_(2), C_(3)` and `C_(4)` are arbitrary is

Options

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

Answer

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

\[ y = (C_1 + C_2) \cos(x + C_3) - C_4 e^{x^4} \]

where \( C_1, C_2, C_3, \) and \( C_4 \) are arbitrary constants, we can follow these steps:

### Step 1: Identify the arbitrary constants in the solution
The given solution contains four arbitrary constants: \( C_1, C_2, C_3, \) and \( C_4 \).

### Step 2: Relate the number of arbitrary constants to the order of the differential equation
In differential equations, the order of the equation is equal to the number of arbitrary constants present in its general solution. This is a standard result in the theory of differential equations.

### Step 3: Count the arbitrary constants
From the solution, we see that there are four arbitrary constants:
- \( C_1 \)
- \( C_2 \)
- \( C_3 \)
- \( C_4 \)

### Step 4: Conclude the order of the differential equation
Since there are four arbitrary constants, the order of the differential equation is 4.

Thus, the order of the differential equation is:

**Order = 4**

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