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Math · Advanced Math
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B The Phase Line ing the direction field of a differential equation dy/di - f(t, y) is particularly easy when the equation is autonomous-that is, the independent variable t does not appear explicitly: dy In Figure 1.18(a) on page 34 the graph exhibits the direction field for with A> 0, and some solutions are sketched. Note the following properties of the graphs and explain how they follow from the fact that the equation is autonomous: (a) The slopes in the direction field are all identical along horizontal lines. (b) New solutions can be generated from old ones by time shifting [i.e., replacing y(t) with From observation (a) it follows that the entire direction field can be described by a single direction line, as in Figure 1.18(b). Of particular interest for autonomous equations are the constant, or equilibrium, solutions y(t)-y, i-l , 2, 3. The equilibrium y yı is called a stable equilibrium, or sink, because the neighboring solutions are attracted to it as t Equilibria that repel neighboring solutions, like у У2, are known as sources; all other equilibria are called nodes, illustrated by y y3-Sources and nodes are unstable equilibria.

Please answer (a) and (b)

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