Differential equation inhomogeneous

In the same way, with similar assumptions, we obtain the (inhomogeneous) differential equation for the components of X13... 

The dependence of the in-phase and quadrature lock-in detected signals on the modulation frequency is considerably more complicated than for the case of monomolecular recombination. The steady state solution to this equation is straightforward, dN/dt = 0 Nss — fG/R, but there is not a general solution N(l) to the inhomogeneous differential equation. Furthermore, the generation rate will vary throughout the sample due to the Gaussian distribution of the pump intensity and absorption by the sample... 

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

Consider first the inhomogeneous differential equation as given by Eq. (5-37). For simplicity, assume here that the oscillator is not damped hence, A = 0. The problems to be treated are now represented by the differential equation... 

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)... 

The most widely used method of solving second-order inhomogeneous differential equations is due to Green. Because it is formal, it is easily adapted to many situations. For the diffusion of an ion-pair under the influence of an interaction energy U, the diffusion equation for the density at r, t, given that the ion-pair was formed at r0, f0, is given by [cf. eqn. (316)]... 

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... 

The careful reader should have realized that we choose not to break up this operator with another Trotter factorization, as was done for the extended system case. In practice, one does not multiple-time-step the modified velocity Verlet algorithm because it will, in general, have a unit Jacobian. Thus, one would like the best representation of the operator that can be obtained in closed form. However, even in the case of a modified velocity Verlet operator that has a nonunit Jacobian, multiple-time-stepping this procedure can be costly because of the multiple force evaluations. Generally, if the integrator is stable without multiple-time-step procedures, avoid them. The solution to this first-order inhomogeneous differential equation is standard and can be found in texts on differential equations (see, e.g.. Ref. 53). 

The salient features of Eq. [188] are that the term in brackets is handled in exactly the same fashion as the bracketed term in Eq. [172]. The terms in curly braces will be factorized to avoid the inhomogeneous differential equation to yield... 

Equation (1) is an inhomogeneous differential equation which can be solved by the method of varying the constanf invented by Lagrange. The corresponding homogeneous equation... 

Equation (8.50) is an inhomogeneous differential equation for (z), whose solution can be written as... 

The substitution x = r 1 P(r) cos 0 gives the second-order inhomogeneous differential equation... 

If one adds to the ip2k, which satisfy (209) to (213), an arbitrary multiple of ipo, they remain solutions of (209) to (213). To make the solutions of the inhomogeneous differential equations of perturbation theory unique, one must impose a normalization condition. Two particular normalizations are convenient. 

The general solution of the inhomogeneous differential equation (4E.5) is given by the general solution of corresponding homogeneous one and a special, non-trivial solution of the inhomogeneous equation. Such special solution is... 

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... 

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