General solution

Definition

A general solution to a differential equation includes all possible solutions and typically contains arbitrary constants. It represents the family of curves that satisfies the differential equation.

5 Must Know Facts For Your Next Test

1. The general solution of a first-order differential equation usually involves one arbitrary constant.
2. For second-order differential equations, the general solution involves two arbitrary constants.
3. To find the general solution, you may need to integrate or use methods such as separation of variables or integrating factors.
4. The general solution can be transformed into a particular solution by applying initial conditions or boundary conditions.
5. In some cases, finding the general solution requires solving a characteristic equation.

Review Questions

• What is included in the general solution of a differential equation?
• How does the number of arbitrary constants in a general solution relate to the order of the differential equation?
• What is one method you can use to find a general solution for a first-order differential equation?

Related terms

Particular Solution:

A specific instance of the general solution obtained by applying initial or boundary conditions.

Homogeneous Differential Equation: A differential equation where every term is dependent on the function and its derivatives.

Characteristic Equation:

An algebraic equation derived from a linear differential equation used to find solutions involving exponential functions.