131F08.classnotes1 - Class notes #1 for MTH G131: Fall 2008...

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Unformatted text preview: Class notes #1 for MTH G131: Fall 2008 Linear first order ODE’s Suppose that p(x), q (x) are continuous in the open interval (a, b), and x0 ∈ (a, b). Then there is a unique solution y = g (x) of the linear ODE dy + p(x)y = q (x), dx a<x<b satisfying g (x0 ) = y0 for any real number y0 . Existence and uniqueness for first order ODE’s Consider the Initial Value Problem dy = f (x, y ), y (x0 ) = y0 dx Suppose f and ∂f are continuous in some rectangle α < x < β , γ < y < δ ∂y containing (x0 , y0 ). Then in some interval x0 − h < x < x0 + h contained in α < x < β , there is a unique solution y = g (x) of the IVP. Solution for first order separable ODE’s This class includes both ‘pure time’ and autonomous first order ODE’s. dy = h(x) f (y ) dx The right side is a product of a function of x (the independent variable) and a function of y (the dependent variable). The formal solution is dy = f (y ) h(x) dx + C ...
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This note was uploaded on 10/19/2011 for the course MATH 3354 taught by Professor Drager during the Fall '08 term at Texas Tech.

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