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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
+ 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
= f (x, y ), y (x0 ) = y0
Suppose f and ∂f are continuous in some rectangle α < x < β , γ < y < δ
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.
= h(x) f (y )
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
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.
- Fall '08