Howtosignalprocessing
- How do you tell if an initial value problem has a unique solution?
- Do initial value problems always have a unique solution?
- How do you know if a function has a unique solution?
- How do you verify that the function is a solution of the initial value problem?
How do you tell if an initial value problem has a unique solution?
How do you know if an initial value problem has a unique solution? that contains a point (xo, yo), then for the initial value problem y' = f(x, y), y(xo) = yohas a unique solution on some open sub-interval of (a, b) which contains the point xo.
Do initial value problems always have a unique solution?
Higher-order equations might have an initial value for both y and y′, for example. Second, an initial value problem doesn't always have a unique solution. It's possible for an initial value problem to have multiple solutions, or even no solution at all.
How do you know if a function has a unique solution?
In a set of linear simultaneous equations, a unique solution exists if and only if, (a) the number of unknowns and the number of equations are equal, (b) all equations are consistent, and (c) there is no linear dependence between any two or more equations, that is, all equations are independent.
How do you verify that the function is a solution of the initial value problem?
Example: Verifying a Solution to an Initial-Value Problem
For a function to satisfy an initial-value problem, it must satisfy both the differential equation and the initial condition. To show that y y satisfies the differential equation, we start by calculating y′ y ′ . This gives y′=−4e−2t+et y ′ = − 4 e − 2 t + e t .
