# Read e-book online Age of Conquerors PDF

By Tim Newark

Книга описывает завоевательские войныот эпохи фараонов и вплоть до 19-го столетия

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When λ1 = λ2 , it is known from elementary linear algebra that the eigenvectors e1 and e2 , corresponding to the eigenvalues λ1 and λ2 , are linearly independent. 1) is given by x(t) = C1 exp(λ1 t)e1 + C2 exp(λ2 t)e2 , where C1 , C2 are constants. In fact, for any given initial condition, this solution is unique by the existence and uniqueness theorem. 48 2. 2). The parabola has equation T 2 − 4D = 0, where D = det(A) and T = trace(A). Definition 5. 2). Then the stable and unstable manifolds of the critical point 0 are defined by ES (0) = {x : + (x) = 0}, where + is the positive limit set and in Chapter 4).

5. 5. 2) all contain straight line trajectories that remain on the coordinate axes forever and exhibit exponential growth or decay along it. These special trajectories are determined by the eigenvectors of the matrix A and are called the manifolds. If the trajectories move towards the critical point at the origin as t → ∞ along the axis, then there is exponential decay and the axis is called a stable manifold . If the trajectories move away from the critical point as t → ∞, then the axis is called an unstable manifold .

The differential equation is separable. Separate the variables and integrate both sides with respect to t. 1. Simple Differential Equations and Applications 19 and so xdx = − tdt. Integration of both sides yields t 2 + x2 = r 2, where r 2 is a constant. There are an infinite number of solutions. The solution curves are concentric circles of radius r centered at the origin. There are an infinite number of solution curves that would fill the plane if they were all plotted. 1. 1: Three solution curves for Example 1.