SECTION 10.7 THE WAVE EQUATION Suppose we have an elastic string (violin string, guitar string, guy wire, electric power line, . . . ) stretched and fastened at its ends x = 0 and x = L. Set this string in motion, by plucking for example, so that it vibra
SECTION 10.5 SEPARATION OF VARIABLES AND THE HEAT EQUATION, AGAIN EXAMPLE. Determine whether the method of separation of variables can be used to replace the following partial dierential equation by a pair of ordinary dierential equations. If so, nd the e
SECTION 10.4 FUNCTIONS ON [0, L] VIA ODD AND EVEN EXTENSIONS Remember that to solve the heat/temperature problem that started all this, we needed to express a function f (x) as a sum of sine functions only, but only on the interval (0, L). So far we know
SECTION 10.3 WHEN DOES A FOURIER SERIES CONVERGE? THE FOURIER CONVERGENCE THEOREM. Suppose that f is originally dened for L x < L and the denition is then extended so that f is periodic with period 2L. Suppose also that then f and f are piecewise continuo
SECTION 10.2 FOURIER SERIES BUT FIRST, THE HEAT EQUATION First, some motivation. Suppose we have a rod of length L, insulated along its sides so heat ows only along the rod. Suppose further that the rod is made of a homogeneous material and has uniform cr
SECTION 10.1 TWO-POINT BOUNDARY VALUE PROBLEMS A typical two-point boundary value problem consists of a dierential equation y + p(x)y + q (x)y = g (x) together with boundary conditions y () = y0 and y ( ) = y1 . Contrary to initial value problems, such bo
SECTION 6.5 IMPULSE FUNCTIONS AND THE DIRAC DELTA FUNCTION EXAMPLES. (1) An electrical circuit is hit by lightning. (2) A subterranean rock layer is hit by an explosion. (3) Alex Rodriguez hits a baseball. (4) A heavy swinging pendulum is hit by a sledgeh
SECTION 6.4 DISCONTINUOUS FORCING FUNCTIONS EXAMPLES. Find the solution of the given initial value problem. Draw the graphs of the solution and of the forcing function. 1. y + 4y = sin t + u (t) sin(t ); y (0) = 0, y (0) = 0
2.
y + y = g (t); y (0) = 0, y
SECTION 6.3 STEP FUNCTIONS AND TRANSLATIONS Because of the applications of the dierential equation y + ay + by = g (t) in mechanical vibration and electrical circuit problems, the function g (t) is sometimes called the forcing function. We need to learn h
SECTION 6.2 USING THE LAPLACE TRANSFORM A CALCULATION. Suppose f (t) does not grow too fast as t and that f (t) is continuous for t 0. Find the Laplace transform of f (t).
MORE CALCULATION. What about the Laplace transform of f (t)? Higher derivatives?
Th
SECTION 6.1 THE LAPLACE TRANSFORM The Laplace transform is a particularly useful example of a general process in mathematics, namely, transform a problem from a dicult setting like calculus into a simpler setting like algebra, solve the transformed proble
SECTION 5.4 SOLUTIONS NEAR SINGULAR POINTS AND EULER EQUATIONS When P (x), Q(x) and R(x) are polynomials, then to nd the singular points of P (x)y + Q(x)y + R(x)y = 0, divide through by P (x), reduce the resulting fractions, and nd the places where the re