The general solution

We now pass to a more general consideration of transport and recombination. Recalling eqn. (352) 

There are, in principle, three zones in which different kinetic laws may operate following Albery and Bartlett [131] we denote these by I—III as in Fig. 60 where 

Region Rate Law Variation of recombination rate with x 

II R = np l ntcxv + Recombination constant assuming a Boltzmann distribution 

Examination of Fig. 69 shows that, within the depletion layer, p first falls and then, near the surface, rises rapidly. It follows from eqn. (435) that the recombination law is likely to remain first order near the depletion layer boundary and, if there is a change of mechanism, it will occur well within the depletion layer. This is an important simplification in two respects. 

For a single particle of mass m moving freely in the absence of any potential energy, the Schrodinger equation becomes  

This energy can be broken down into three independent parts corresponding to the velocity components Uy and along the three axes  

Assume now that Y can be written as the product of three independent functions, each containing only one independent variable, 

It can be easily shown that introduction of Eqs 16.4.2 and 16.4.3 into Eq. 16.4.1 leads to three easy to solve differential equations, each containing only one ind endent variable (Problem 16.4)  

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By substituting relations (26) into equations (24), (25) we obtain the general solution of the equilibrium equations... 

Since in this case the Flamiltonian is time independent, the general solution can be written as... 

With time independent matrix K it has the general solution... 

The symmetric transmission coefficients are defined = LijMi. The general solutions are of the form... 

The general solutions for xi and 31 2 are superpositions, that is, linear combinations of all of the solutions we have found... 

The general solution to the radial equation is then taken to be of the form ... 

Perform a separation of variables and indieate the general solution for the following expressions ... 

Taking r to be held constant, write down the general solution, ((1)), to this Schrodinger... 

The general solution to this equation is the now familiar expression ... 

In this case, Oq is the maximum amplitude of the stress. The solution to this differential equation will give a functional description of the strain in this dynamic experiment. In the following example, we examine the general solution to this differential equation. 

Equations (3.77) and (3.81) both have the same general form dy/dt + Py = Q, so the general solution-given in Example 3.5—is the same for both, although the values of the constants are different. When the constants are evaluated, the storage and loss components of the modulus are found to be... 

A relation between the variables, involving no derivatives, is called a solution of the differential equation if this relation, when substituted in the equation, satisfies the equation. A solution of an ordinaiy differential equation which includes the maximum possible number of arbitrary constants is called the general solution. The maximum number of arbitrai y constants is exactly equal to the order of the dif-... 

In the case of some equations still other solutions exist called singular solutions. A singular solution is any solution of the differential equation which is not included in the general solution. 

A solution of a difference equation is a relation between the variables which satisfies the equation. If the difference equation is of order n, the general solution involves n arbitraty constants. The techniques for solving difference equations resemble techniques used for differential equations. 

Factorization If the difference equation can be factored, then the general solution can be obtained by solving two or more successive equations of lower order. Consider yx 2 + A y -1- = ( )(x). If there... 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Using the isotherm to calculate loadings in equilibrium with the feed gives rii = 3.87 mol/kg and ri2 = 1.94 mol/kg. An attempt to find a simple wave solution for this problem fails because of the favorable isotherms (see the next example for the general solution method). To obtain the two shocks, Eq. (16-136) is written... 

The general solution of this second order differential equation is... 

Since computer cahinets stand on the floor, the general solution is to hlow the cold air up from a false floor directly into the cahinets, with a lesser volume being hlown into the room to deal with other heat loads. The air-conditioning unit will now stand on the floor, taking warm air from the upper part of the room and blowing it down into the false floor (see Figure 28.10). 

For the purposes of fixing the stationary states we have up to this point only considered simply or multiply periodic systems. However the general solution of the equations frequently yield motions of a more complicated character. In such a case the considerations previously discussed are not consistent with the existence and stability of stationary states whose energy is fixed with the same exactness as in multiply periodic systems. But now in order to give an account of the properties of the elements, we are forced to assume that the atoms, in the absence of external forces at any rate always possess sharp stationary states, although the general solution of the equations of motion for the atoms with several electrons exhibits no simple periodic properties of the type mentioned (Bohr [1923]). 

Tenets (i) and (ii). These are applicable only where the reactant undergoes no melting and no systematic change of composition (e.g. by the diffusive removal of a constituent) and any residual solid product phase offers no significant barrier to contact between reactants or the escape of volatile products [33,34]. When all these conditions are obeyed, the shape of the fraction decomposed (a) against time (f) curve for an isothermal reaction can, in principle, be related to the geometry of formation and advance of the reaction interface. The general solution of this problem involves intractable mathematical difficulties but simplifications have been made for many specific applications [1,28—31,35]. 

In fact, such a method was proposed by Sack in the classical work [99], which was far ahead of its time. This method provides the general solution of Eq. (6.4) in the form of a continuous fraction, which is, however, rather difficult to analyse. In the case of weak collisions, there is no good alternative to this method, but for strong collisions, the solution can be found analytically. Let us first consider this case. 

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