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 odes 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 odes 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 ( x, y ) such that d dx = M ( x, y ) + N ( x, y ) y . Criterion for exactness Assume M, N, M y , N x are continuous in the rectangle R : < x < , < y <...
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