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Additional resources for Applied Mathematics for Physical Chemistry (2nd Edition)
Consider, first, the reduced equation, where R(x) = 0. and y = e-" are each solutions to Equation (6-l4), calledparticular solutions. A general solution to Equation (6-14), when the roots to the auxiliary equation are real, is found by taking a linear combination of the two particular solutions. where c, and c, are arbitrary constants. We see, then, that when the roots to the auxiliary equation are real, the solution to the differential equation is real and is the combination of an exponential increase plus an exponential decay.
The first approach allows us to mathematically generate integrals. The second approach allows us to assign a physical meaning to the integral. Introductory courses on integral calculus spend a tremendous amount of time on the first approach, teaching all the various methods for generating integrals. While this is important, and perhaps at some point should be learned, in practice it is rarely used, most of us referring to tables of integrals to do our integrating. Thus, in this text we shall emphasize using tables of integrals for performing the mechanics of integration.
Consider the equation which has the real, double roots m = 2, 2. The general solution to the equation is y = clek + c2xe2 + d2y 5 y = 0. 2. Solve dx2 Substituting y = em and its derivatives into this equation gives which has the imaginary roots m = hi&. The general solution to the equation is y = c1ei& + c2e-iJSx where 0 is a constant. We easily can show, using the method outlined in Section 6-4, that two particular solutions of Equation (6-22) are sin px and cos px. Let us assume, however, that the solution to Equation (6-22) is a series of the form where K represents the lowest power thatx can have in the summation.