math4b-8 - Math 4B Lecture 8 Doug Moore Announcements 1...

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Math 4B Lecture 8 April 24, 2013 Doug Moore
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Announcements. 1. Midterm next Wednesday. You should know linear DE’s (dif- ferential equations), exact differentials, Cauchy-Euler poly- gon and homogeneous linear second order DE’s with constant coefficients. No calculators, computers, notes or books al- lowed. 2. Homework 5 is short and will be due next Wednesday, May 1 at 6pm (the day of the midterm).
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Many of the differential equations encountered in physics and en- gineering involve acceleration—the second derivative of position— and are therefore of second order. The standard form for a second order differential equation is d 2 y dt 2 = f ( t, y, dy dt ) , (1) where f is a suitably well-behaved function. For example, d 2 y dt 2 = sin dy dt 2 + ty ! is in standard form. Just as for first order differential equations, there is a fundamental existence and uniqueness theorem for second order equations which are in standard form.
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Picard’s Theorem. If f is continuous and has continuous first partial derivatives, then the differential equation (1) has a unique differentiable solution which satisfies the initial conditions x ( t 0 ) = c 0 , dx dt ( t 0 ) = c 1 . Thus the general solution to a second order differential equation possesses two constants of integration, which are often deter- mined by initial conditions. Just as for first order differential equations, there are no completely general methods for solving second order differential equations. Nevertheless, there are some useful methods which work in special cases. One such method is the method of reduction of order . For an example, let us consider the equation of a cart running along a friction-free track and attached to a wall by means of a spring.
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. Let x ( t ) denote the directed distance from equilibrium to the position of the cart at time t . There are two expressions for the force F which acts on the cart. The first of these is given by Hooke’s spring law, F = - kx, where k is a positive constant, called the spring constant . The
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