M 427K - Definitions and Theorems

M 427K - Definitions and Theorems - M427k Some Definitions...

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Unformatted text preview: M427k: Some Definitions and Theorems • Existence and uniqueness for first order linear ode’s If a ( x ) and b ( x ) are continuous on some open interval containing x , and c is any given number, then there is one and only one solution in the interval to the system y ( x ) + a ( x ) y + b ( x ) = 0; y ( x ) = c . • Existence and uniqueness for general first order ode’s Consider y = f ( x, y ); y ( x ) = y . Assume f and ∂f ∂y are continuous in some rectangle R : | x- x | ≤ A, | y- y | ≤ B . Let M = max x,y ∈ R | f ( x, y ) | and c = min( A, B M ). Then there is one and only one solution y ( x ) valid for x- c < x < x + c . • A first order ode is “separable” if it can be put in the form f ( y ) y = g ( x ). • The ode M ( x, y ) + N ( x, y ) y = 0 is “exact” if there is some function φ ( x, y ) such that dφ dx = M ( x, y ) + N ( x, y ) y . • Criterion for exactness Assume M, N, ∂M ∂y , ∂N ∂x are continuous in the rectangle R : α < x < β, γ < y < δ...
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