Initial value

Definition

The initial value is the starting point of a solution to a differential equation, often given as the value of the function and its derivatives at a specific point. It is essential for determining the unique solution of an initial value problem.

5 Must Know Facts For Your Next Test

1. Initial values are necessary to solve initial value problems (IVPs) in differential equations.
2. Without an initial value, the differential equation might have infinitely many solutions.
3. The existence and uniqueness theorem states that if certain conditions are met, there is exactly one solution to the IVP near the initial value.
4. Initial values can be specified for higher-order derivatives as well.
5. Finding a particular solution involves substituting the initial values into the general solution of the differential equation.

Review Questions

• Why are initial values important for solving differential equations?
• How does the existence and uniqueness theorem relate to initial values?
• What happens if no initial value is provided for a differential equation?

Related terms

Differential Equation: An equation involving derivatives of a function or functions.

Initial Value Problem (IVP): A differential equation coupled with a specific value at a starting point, used to find a unique solution.

Existence and Uniqueness Theorem: A theorem that provides conditions under which an IVP has exactly one solution in some interval around the initial value.