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By Calvin H. Wilcox

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Y − y = 0 17. y + y − 6y = 0 18. y − 3y + 2y = 0 In each of Problems 19 and 20 determine the values of r for which the given differential equation has solutions of the form y = t r for t > 0. 19. t 2 y + 4ty + 2y = 0 20. t 2 y − 4ty + 4y = 0 In each of Problems 21 through 24 determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear. Partial derivatives are denoted by subscripts. 21. uxx + uyy + uzz = 0 22. uxx + uyy + uux + uuy + u = 0 23.

September 11, 2008 11:18 boyce-9e-bvp Sheet number 46 Page number 26 26 cyan black Chapter 1. Introduction (b) Show that the potential energy V of the pendulum, relative to its rest position, is V = mgL(1 − cos θ). (c) By the principle of conservation of energy, the total energy E = T + V is constant. Calculate dE/dt, set it equal to zero, and show that the resulting equation reduces to Eq. (12). 31. A third derivation of the pendulum equation depends on the principle of angular momentum: the rate of change of angular momentum about any point is equal to the net external moment about the same point.

Another aspect of computer use that is very relevant to the study of differential equations is the availability of extremely powerful and general software packages that can perform a wide variety of mathematical operations. Among these are Maple, Mathematica, and MATLAB, each of which can be used on various kinds of personal computers or workstations. All three of these packages can execute extensive numerical computations and have versatile graphical facilities. Maple and Mathematica also have very extensive analytical capabilities.

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