Consider the problem alpha^2 u_xx = u_t, 0 0; u(x, 0) = f(x), 0 LE x LE L. (i) L

Question

Consider the problem alpha^2 u_xx = u_t, 0 0; u(x, 0) = f(x), 0 LE x LE L. (i) Let u(x, t) = X(x)T(t),and show that x" + lambda X = 0, X(0) = 0, x'(L) + gamma X(L) = 0, (ii) and T' + lambda alpha^2 T = 0, where lambda is the separation constant. Assume that lambda is real, and show that problem (ii) has no nontrivial solutions if lambda LE 0. If lambda > 0, let lambda. = mu^2 with mu > 0. Show that problem (ii) has nontrivial solutions only if mu is a solution of the equation mu cos mu L + gamma sin mu L = 0.(iii)

Explanation / Answer

The solution of the equation for simple harmonic oscillations may be expressed in terms of trigonometric functions. A common problem in physics is to determine the motion of a particle in a given force field. For a particle moving on a line, the force field is given by specifying the force F as a function of the position x and time t. The problem is to write x as a function of the time t so that the equation d2x F = m - (Force = Mass X Acceleration) dt2 (1) is satisfied, where m is the mass of the particle. Equation (1) is called Newton's second law of motion.' If the dependence of F on x and t is given, equation (1) becomes a differential equation in x-that is, an equation involving x and its derivatives with respect to t. It is called second-order since the second derivative of x appears. (If the second derivative of x were replaced by the first derivative, we would obtain a first-order differential equation-these are studied in the following sections). A solution of equation (1) is a function x = f(t) which satisfies equation (1) for all t when f(t) is substituted for x. ' Newton always expressed his laws of motion in words. The first one to formulate Newton's laws carefully as differential equations was L. Euler around 1750. (See C. Truesdell, Essays on the History of Mechanics, Springer-Verlag, 1968.) In what follows we shall not be concerned with specific units of measure for force--often it is measured in newtons (1 newton = 1 kilogram-meter per second2). Later, in Section 9.5, we shall pay a little more attention to units.

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