Existence and Uniqueness Theorem
a linear system is consistent if and only if the rightmost column of the augmented matrix is NOT a pivot column – that is, if and only if an echelon form of the augmented matrix has NO row of the form [0 … 0 b] with b nonzero. If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable.
Existence and Uniqueness Theorem is a fundamental theorem in mathematics that establishes the conditions under which a solution exists and is unique for certain types of differential equations. The theorem states that for a first-order differential equation with a given initial condition, there exists a unique solution in some interval of the independent variable, provided that the function and its partial derivative are continuous in that interval.
The Existence and Uniqueness Theorem applies to both ordinary differential equations and partial differential equations. It guarantees the existence of a solution, and furthermore, that solution is unique for the given initial condition. This is a very powerful theorem and is used extensively in various fields of mathematics, physics, engineering, and economics.
Theorem Statement: For an ordinary differential equation of the form y’ = f(x,y) that satisfies the following conditions:
1. f(x,y) is continuous in some rectangle D = { (x,y) : a < x < b, c < y < d }
2. f(x,y) satisfies a Lipschitz condition with respect to y, i.e., there exists a constant L > 0 such that |f(x,y1) – f(x,y2)| ≤ L|y1 – y2| for all (x,y1), (x,y2) ∈ D
Then, given a point (x0, y0) ∈ D, there exists a unique solution y(x) to the initial value problem y’ = f(x,y), y(x0) = y0 defined on some interval containing x0.
In summary, the Existence and Uniqueness Theorem is a crucial theorem that helps us determine whether a solution exists and is unique for a given differential equation under certain conditions.
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