key term - Particular solution

Definition

A particular solution of a differential equation is a specific solution that satisfies both the differential equation and any given initial or boundary conditions. It is distinct from the general solution, which includes a family of solutions with arbitrary constants.

5 Must Know Facts For Your Next Test

1. A particular solution is obtained by applying initial or boundary conditions to the general solution of a differential equation.
2. Particular solutions are unique for given initial or boundary conditions.
3. To find a particular solution, you first solve the differential equation to obtain the general solution, then substitute the initial/boundary conditions to find specific values for the constants.
4. Inhomogeneous differential equations require finding both the homogeneous and particular solutions to form the complete solution.
5. Methods such as undetermined coefficients or variation of parameters are often used to find particular solutions for non-homogeneous linear differential equations.

Review Questions

Related terms

General Solution:

The family of all possible solutions to a differential equation, typically containing arbitrary constants.

Homogeneous Differential Equation: A type of differential equation where every term is dependent on the function and its derivatives, with no independent terms.

Initial Condition:

\text{Values provided for } \text{a function and its derivatives at a specific point, used to find} \text{a particular solution.}