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existence and uniqueness theorem for (1.1) we just have to establish that the equation (3.1) has a unique solution in [x0 −h,x0 +h]. IV. Proof of the uniqueness part of the theorem. Here we show that the problem (3.1) (and thus (1,1)) has at most one solution (we have not yet proved that it has a solution at all).

The following constitute the Existence and Uniqueness Theorems from the text: Existence Theorem: If f(t;x) is de ned and continuous on a rectangle R in the tx-plane, then given any point (t 0;x 0) 2R, the initial value problem x0= f(t;x)andx(t 0)=x 0 has a solution x(t) de ned in an interval containing t 0. Uniqueness Theorem: If f(t;x) and @f @x

existence and uniqueness theorem for (1.1) we just have to establish that the equation (3.1) has a unique solution in [x0 −h,x0 +h]. IV. Proof of the uniqueness part of the theorem. Here we show that the problem (3.1) (and thus (1,1)) has at most one solution (we have not yet proved that it has a solution at all).

In terms of the Existence and Uniqueness Theorem, we have f ( x, y) = − P ( x) y + Q ( x) and ∂ f ∂ y ( x, y) = − P ( x). But both P ( x) and Q ( x) are assumed continuous on the rectangle R = { ( x, y) | a ≤ x ≤ b, c ≤ y ≤ d } for any values of c and d, and f ( x, y) is a combination of continuous functions.

Existence Uniqueness Theorem We will now see that rather mild conditions on the right hand side of an ordi-nary di erential equation give us local existence and uniqueness of solutions. De nition. Let f: D!Rnbe a coninuous function de ned in the open set D Rn+1. We say that f is locally Lipschitz in the Rn variable if for each (t 0;x

Theorem (Existence and Uniqueness Theorem for ODE). Suppose f(t;x) is continuous in the open set D Rn+1 and is locally Lipschitz in xin D. Let (t 0;x 0) 2D. Then, the initial value problem x_ = f(t;x);x(t 0) = x 0 (1) has a unique solution de ned in a small interval Iabout t 0 in R. Proof. Let Ube an open neighborhood about (t 0;x 0) in Dso that

Existence and Uniqueness Theorem. The system Ax = b has a solution if and only if rank (A) = rank(A, b). The solution is unique if and ...

(a) is an existence theorem. It guarantees that a solution exists on some open interval that ...

Theorem 1 (Existence). Suppose that F(x;y) is a continuous function de ned in some region R = f(x;y) : x0 < x < x0 + ;y0 < y < y0 + g containing the point (x0;y0). Then there exists a number 1 (possibly smaller than ) so that a solution y = f(x) to (*) is de ned for x0 1 < x < x0 + 1. Theorem 2 (Uniqueness). Suppose that both F(x;y) and @F

Picard’s Existence and Uniqueness Theorem Denise Gutermuth These notes on the proof of Picard’s Theorem follow the text Fundamentals of Di↵erential Equations and Boundary Value Problems, 3rd edition, by Nagle, Sa↵, and Snider, Chapter 13, Sections 1 and 2.

Theorem 2 (Uniqueness). Suppose that both F(x;y) and @F @y (x;y) are continuous functions de ned on a re-gion R as in Theorem 1. Then there exists a number 2 (possibly smaller than 1) so that the solution y = f(x) to (*), whose existence was guaranteed by Theorem 1, is the unique solution to (*) for x0 2 < x < x0 + 2. x − 0 δ 2 x + 0 δ 2 0 ...

NOTES ON THE EXISTENCE AND UNIQUENESS THEOREM. FOR FIRST ORDER DIFFERENTIAL EQUATIONS. I. Statement of the theorem. We consider the initial value problem.

In terms of the Existence and Uniqueness Theorem, we have. f ( x, y) = − P ( x) y + Q ( x) and ∂ f ∂ y ( x, y) = − P ( x). But both P ( x) and Q ( x) are assumed continuous on the rectangle R = { ( x, y) | a ≤ x ≤ b, c ≤ y ≤ d } for any values of c and d, and f ( x, y) is a combination of continuous functions.

theorem on existence and uniqueness of first order ODE (with initial value), basically, under a very simple (easily verified) condition (which is a strong ...

Similarly, an ODE may also have no solution, a unique solution or infinitely many solutions. The existence theorem is used to check whether there exists a ...

(ii) Under what conditions can we be sure that there is a unique solution to (*)? Here are the answers. Theorem 1 (Existence). Suppose that F(x, y) is a.

The Picard–Lindelöf theorem shows that the solution exists and that it is unique. The Peano existence theorem shows only ...