By Isaac Todhunter

This Elibron Classics ebook is a facsimile reprint of a 1864 variation by way of Macmillan and Co., Cambridge and London.

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**Extra resources for A Treatise on the Differential Calculus: With Numerous Examples**

**Sample text**

Berry and Houston [6] have also given a schematic diagram of a modeling cycle. 4) [21]. The nodes of the diagram represent information to be collected, sorted, evaluated, and organized. The edges of the diagram represent activities of two-way communication between the nodes and the corresponding source of information. S: Problem statement (arising out of real-world situations). M: Mathematical model (uses concepts/variables, relations, restrictions, defines a goal and decides priorities/quality assignments).

14 Show that x(t) = 0, t ≥ t0, is an unstable solution of the equation x + x sin 2x − x = 0. 27 Introduction to Mathematical Modeling SOLUTION The equivalent system is x1 = x2 , x2 = x1 − x2 sin 2x1 , where x = x1. Choose, V(x1, x2) = x1x2. In every neighborhood of the origin, V(x1, x2) takes both positive and negative values. We obtain V *( x) = x12 + x22 − x1x2 sin(2x1 ). In any neighborhood of the origin, |x1|, |x2| are small and sin(2x1) ≈ 2x1. The third term on the right-hand side is of order of one higher than the first two terms.

Then, V *( x) = 2a x1 f1 + 2bx2 f 2 = 2a x1[− x2 − (1/2)x1x22 ] + 2bx2 [3 x1 − (1/3)x2 ] = 2x1x2 [− a + 3b] − ax12 x22 − (2/3)bx22 . 42 Introduction to Mathematical Modeling and Chaotic Dynamics Set 3b − a = 0. We may choose a = 3, b = 1. We get V ( x) = 3 x12 + x22 and V *( x) = − x22 (3 x12 + 2). Now, V * (x) ≤ 0 at all points (x1, x2) ∈ R 2 and V(x1, x2) → ∞ as x12 + x22 → ∞. Hence, all the solutions are bounded as t → ∞. Also, V * (x) = 0 when x2 = 0, that is, at all points on the x1 axis. E is the set of all points on the x1 axis.