The numbers of arbitrary constants in the particular solution of a differential equation of third order are :

Options

A. 3
B. 2
C. 1
D. 0

Answer

To determine the number of arbitrary constants in the particular solution of a third-order differential equation, we can follow these steps:

### Step 1: Understand the Order of the Differential Equation
The order of a differential equation is defined as the highest derivative present in the equation. For a third-order differential equation, the highest derivative is the third derivative.

**Hint:** The order of the differential equation indicates how many times the function has been differentiated.

### Step 2: General Solution and Arbitrary Constants
For a differential equation of order \( n \), the general solution will contain \( n \) arbitrary constants. Therefore, for a third-order differential equation, the general solution will have 3 arbitrary constants.

**Hint:** The general solution of a differential equation includes constants that represent the family of solutions.

### Step 3: Distinguish Between General and Particular Solutions
The general solution includes all possible solutions and contains arbitrary constants. However, a particular solution is derived from the general solution by applying specific initial or boundary conditions, which eliminates some of the arbitrary constants.

**Hint:** A particular solution is specific and does not contain all arbitrary constants.

### Step 4: Determine the Number of Arbitrary Constants in the Particular Solution
In the case of a particular solution, there is only one arbitrary constant remaining, regardless of the order of the differential equation. This is because the particular solution is obtained by fixing the values of the constants in the general solution.

**Hint:** A particular solution is unique and typically has only one arbitrary constant.

### Conclusion
Therefore, the number of arbitrary constants in the particular solution of a third-order differential equation is **1**.

**Final Answer:** 1