Linear inhomogeneous differential equations

The First-Order Linear Inhomogeneous Differential Equation (FOLIDE) First-Order Reaction Including Back Reaction Reaction of Higher Order Catalyzed Reactions... 

In Eqs. 12-47/48 we recognize the first-order linear inhomogeneous differential equation (FOLIDE, Box 12.1). Depending on whether the input I and the different rate constants (kw, kj) are constant with time, their solutions are given in Eqs. 6,8, or 9 of Box 12.1. 

Box21.6 Solution of Two Coupled First-Order Linear Inhomogeneous Differential Equations (Coupled FOLIDEs)... 

Equations (2.8) and (2.9) describe the time development of the number of nuclei of any isotope i in a radioactive decay series by means of n coupled linear inhomogeneous differential equations. The general solution of any of these equations is the summation of the general solution of the homogeneous equation... 

As is known from the differential equations theory, the Green functions are used for solution of the linear inhomogeneous differential equations ... 

Poisson s eqtiation Second-order linear inhomogeneous differential equation of electric potential or other electrical or magnetic functions. 

Unfortunately, obtained equation could not be solved by means of Mathcad symbolic core directly . We definitely can differentiate it with respect to variable t and get linear inhomogeneous differential equation of second-order with respect to derivative. After that we can use the methodic of getting partial solution, given nearly in every handbook of differential equations. However it will mainly be a hand work, and not a computational calculation. Symbolic resources of Maple allow finding the solution of its equation directly and getting analytic expressimi for time-dependence of intermediate s concentration (Fig. 1-9) ... 

We have obtained an inhomogeneous differential equation with constant coefficients. As follows from the theory of linear equations, its solution is a sum... 

An inhomogeneous differential equation contains a term that is not proportional to the unknown function or to any of its derivatives. An example of a linear inhomogeneous equation is... 

The model of isotope transfer in CSTR is a set of ordinary inhomogeneous differential equations with linear right side, which are solved by standard methods of expansion into eigenvectors. With the known eigenvalues and eigenvectors, it is easy to calculate first- and second-order derivatives, which makes the minimization procedure much faster. 

Since the concentrations will decay monotonically towards their equilibrium values, if possible, it can be asserted that they will also decay monotonically towards a stationary state, when the stationary state is inside the linear domain. This is so, because the inhomogeneous differential equation which describes the approach to the stationary state has (15) as its homogeneous part. As there is only one stationary state of the system corresponding to a given set of fixed forces or fluxes, and this state is a state of minimum entropy production, we have seen that the system in time approaches a state of minimum entropy production. This is sometimes formulated as a direct consequence of the existence of a state of minimum entropy production, but it is clear that in that caise one alwa assumes in addition to the existence of the minimum a relationship between a and d such as (15), since only then do the equations give a temporal description of the system. 

This is an inhomogeneous linear differential equation of second order with constant coefficient a, where g is its right hand side. The parameter a is very small, and it is approximately... 

If J(t) from Eq. 21-11 or 21-12 is inserted into Eq. 21-4, we get a linear differential equation with a time variable inhomogeneous term but constant rate k. The corresponding solution is given in Box 12.1, Eq. 8. Application of the general solution to the above case is described in Box 21.3. The reader who is not interested in the mathematics can skip the details but should take a moment to digest the message which summarizes our analytical exercise. 

I. Linear differential equations in which only the inhomogeneous term is a random function, such as the Langevin equation. Such equations have been called additive and can be solved in principle. 

Since the coefficients of each member in the set of differential equations are spatially dependent and the equations themselves inhomogeneous, higher order terms are obtained with increasing difficulty. This is in contrast to the case with solving the linear PB equation with the same undulating surface. In this case the set member differential equations are simply homogeneous and have constant coefficients,... 

Let us, for a moment, consider a single particle in one dimension with a Hamiltonian of the type// = p2/2m + V(x). This is a second-order differential operator, and this means that the general solution to the inhomogeneous Eq. (3.51)—considered as a second-order differential equation—will consist of a linear superposition of two special solutions, where the coefficients will depend on the boundary conditions introduced. As a specific example, one could think of the two solutions to the JWKB problem, their connection formulas, and the Stoke s phenomenon for the coefficients. 

S.2 Inhomogeneous Linear Differential Equations with Constant Coefficients... 

Consider an inhomogeneous linear differential equation with constant coefficients given by... 

As we have already noted, the eigenvalues for the linearized deterministic system at the inhomogeneous fixed point, plotted as black dots in Figs. 5.25b,c are computed numerically for the system (5.30) in the deterministic case by using the spatially discretized set of ordinary differential equations. 

An inhomogeneous linear differential equation can be solved if a particular solution can be found. 

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