Particular Solution in Differential Equations:
When solving a differential equation, the general solution typically contains arbitrary constants that need to be determined. A particular solution is then found by substituting specific values into these arbitrary constants to satisfy any initial or boundary conditions given in the problem.

Importance of Particular Solution:
The particular solution is crucial as it provides the specific solution that meets the given conditions, making it unique among the infinite solutions that may exist for the differential equation. It allows us to find the exact solution that fits the problem at hand.

Process of Finding Particular Solution:
1. Begin by solving the differential equation to find the general solution, which includes arbitrary constants.
2. Use the initial or boundary conditions provided in the problem to determine the values of these arbitrary constants.
3. Substitute these specific values back into the general solution to obtain the particular solution that satisfies the given conditions.

Verification of Particular Solution:
Once the particular solution is found, it should be verified by substituting it back into the original differential equation to ensure that it indeed satisfies the equation. This step confirms that the particular solution is correct and meets all the requirements of the problem.

Conclusion:
In conclusion, a particular solution for a differential equation is derived by substituting particular values into the arbitrary constants in the general solution. It plays a vital role in providing the unique solution that meets the given conditions of the problem, and its accuracy should always be verified through the verification process.