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L1D - Fall 2003 Math 308/501502 1 Introduction 1.D...

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Fall 2003 Math 308/501–502 1 Introduction 1.D Autonomous Equations, Stability, and the Phase Line Fri, 05/Sep c 2003, Art Belmonte Summary Autonomous Equations A first-order autonomous differential equation is one of the form dy / dt = y 0 = f ( y ) . Notice that the derivative expression does not depend on the independent variable t . Accordingly, along a given horizontal line in the ty -plane (say y = b ), the slopes are the same. Moreover, tangent line segments in the direction field along the vertical line t = a are replicated along any other vertical line t = a + k . Not only does this give the direction field a rather uniform appearance, it also means that the DE lends itself readily to qualitative analysis of solution curves. That is, without even analytically solving the DE (if this is even possible), we can still say a lot about the behavior of its solutions. Equilibria To do this, first find the zeros of f ; i.e., the values y for which f ( y ) = 0. If f ( c ) = 0, then the constant function y c is a solution of the DE, since y 0 0 = f ( c ) = f ( y ( t )) for all t . The value c is called an equilibrium point and the constant function y c is called an equilibrium solution . Looking at a
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