General solution of the differential equations

39) a general equation has been given which allows the calculation of eigenvalues of the Jacobi matrix (that means reaction constants or their combinations) taking the degree of advancement. In the case of consecutive reactions, special solutions have been given. However, this system of differential equations has a general solution [17a], 

Survey on relationships for some consecutive reactions, applications of the formula given in Section 2.2.1.1, further details are given in Appendix 6.5.  

This determinant (eq. (2.46)) is called the characteristic polynomial of the Jacobi matrix. Its rank takes on the degree s. The determinant is explicitly given by 

The derived relationships between these coefficients are of great impor- 

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As discussed previously, the general solution of the differential equation is ... 

The general solution of the differential equations describing this process is complex, and often of limited value in practice, so that it is very difficult to obtain the values of the fundamental rate constants k, and k, . However, one may simplify... 

To obtain the interesting coefficients p., the general solution of the differential equation, eq. (2.45), is differentiated for each degree of advancement s - 1 times. Thus for the kth degree of advancement at time r = 0, the following system of linear equations is obtained [16] ... 

The general solution of the differential equation (25) is a linear combination of the linearly independent solutions, where the constants of combination are determined by the initial conditions. In the special case considered below, from three to five terms of the asymptotic expansions in (26) and (27) are needed to compute fluxes to an accuracy of four decimal places. 

When a second-order differential equation is linear (i.e., does not contain nonlinear terms such as uf du/dxY, etc.), the following important superposition principle applies Any two independent solutions that satisfy the equation and its boundary conditions can be added to obtain the general solution of the differential equation. 

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 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. 

One of the popular branches of modern mathematics is the theory of difference schemes for the numerical solution of the differential equations of mathematical physics. Difference schemes are also widely used in the general theory of differential equations as an apparatus available for proving existence theorems and investigating the differential properties of solutions. 

If the dependent variable y(jt) and all of its derivatives occur in the first degree and do not appear as products, the equation is said to be linear. In effect, the solution of a differential equation of order n necessitates n integrations, each of which involves an arbitrary constant. However, in some cases one or more of these constants may be assigned specific values. The results, which are also solutions of the differential equation, are referred to as particular solutions. The general solution, however, includes all of the n constants of integration, whose evaluation requires additional information associated with the application. 

Equations (5.1) define a direction vector at each point (t,y) of the n+1 dimensional space. Fig. 5.1 shows the field of such vectors for the radioactive decay model (5.2). Any function y(t), tangential to these vectors, satisfies (5.2) and is a solution of the differential equation. The family of such curves is the so called general solution. For (5.2) the general solution is given by... 

It should be noted that, in most chemical situations, we rarely need the general solution of a differential equation associated with a particular property, because one (or more) boundary conditions will almost invariably be defined by the problem at hand and must be obeyed. For example ... 

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