Homogeneous differential equation

The procedure we followed in the previous section was to take a pair of coupled equations, Eqs. (5-6) or (5-17) and express their solutions as a sum and difference, that is, as linear combinations. (Don t forget that the sum or difference of solutions of a linear homogeneous differential equation with constant coefficients is also a solution of the equation.) This recasts the original equations in the foiin of uncoupled equations. To show this, take the sum and difference of Eqs. (5-21),... 

It is a property of linear, homogeneous differential equations, of which the Schroedinger equation is one. that a solution multiplied by a constant is a solution and a solution added to or subtracted from a solution is also a solution. If the solutions Pi and p2 in Eq. set (6-13) were exact molecular orbitals, id v would also be exact. Orbitals p[ and p2 are not exact molecular orbitals they are exact atomic orbitals therefore. j is not exact for the ethylene molecule. 

Linear Differential Equations with Constant Coeffieients and Ri ht-Hand Member Zero (Homogeneous) The solution of y" + ay + by = 0 depends upon the nature of the roots of the characteristic equation nr + am + b = 0 obtained by substituting the trial solution y = in the equation. 

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

The solution now has the form sin (n-Kxll)(Ae + Be ). Since V(a., cc) = 0, A must be taken to be zero because e becomes arbitrarily large as y The solution then reads B sin (m x/l)e where is the multiplicative constant. The differential equation is linear and homogeneous so that 2,r=i sin (nltx/l) is also a solution. Satisfaction of the last boundary condition is ensured by taking... 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

The solution to this fourth-order partial differential equation and associated homogeneous boundary conditions is just as simple as the analogous deflection problem in Section 5.3.1. The boundary conditions are satisfied by the variation in lateral displacement (for plates, 5w actually is the physical buckle displacement because w = 0 in the membrane prebuckling state however, 5u and 8v are variations from a nontrivial equilibrium state. Hence, we retain the more rigorous variational notation consistently) ... 

Thus the kinetic equation may be derived for operator (7.21), though it does not exist for an average dipole moment. Formally, the equation is quite identical to the homogeneous differential equation of the impact theory with the collisional operator (7.27). It is of importance that this equation holds for collisions of arbitrary strength, i.e. at any angle of the field reorientation. From Eq. (7.10) and Eq. (7.20) it is clear that the shape of the IR spectrum... 

Number of k-fold Precursor Particles. Dynamic differential equations were written for the concentration of the k-fold precursors to account for birth and death by coagulation, growth by propagation, and the formation of primary precursors by homogeneous nucleation. There... 

In the theory of difference schemes with a primary family of schemes the coefficients of a homogeneous difference scheme are expressed through the coefficients of the initial differential equation by means of the so-called pattern functionals the arbitrariness in the choice of these functionals is limited by the requirements of approximation, solvability, etc. There are various ways of taking care of these restrictions. The availability of a primary family of homogeneous difference schemes is ensured by a family of admissible pattern functionals known in advance. 

As a result, a considerable amount of effort has been expended in designing various methods for providing difference approximations of differential equations. The simplest and, in a certain sense, natural method is connected with selecting a, suitable pattern and imposing on this pattern a difference equation with undetermined coefficients which may depend on nodal points and step. Requirements of solvability and approximation of a certain order cause some limitations on a proper choice of coefficients. However, those constraints are rather mild and we get an infinite set (for instance, a multi-parameter family) of schemes. There is some consensus of opinion that this is acceptable if we wish to get more and more properties of schemes such as homogeneity, conservatism, etc., leaving us with narrower classes of admissible schemes. 

Stability of difference schemes with respect to coefficients. In solving some or other problems for a differential equation it may happen that coefficients of the equation are specified not exactly, but with some error because they may be determined by means of some computational algorithms or physical measurements, etc. Coefficients of a homogeneous difference scheme are functionals of coefficients of the relevant differential equation. An error in determining coefficients of a scheme may be caused by various... 

We first illustrate its employment for a differential equation in tackling problem (I) with homogeneous boundary conditions... 

Chapters 2-5 are concerned with concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which we try to solve the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified (in such a case the availability of the family is provided by pattern functionals) and schemes of a desired quality should be selected within the primary family. This problem is solved in Chapter 3 using a particular form of the scheme and its solution leads us to conservative homogeneous schemes. 

By introducing a new variable rj r) = rC(r) Eq. (27) transforms into a homogeneous second-order differential equation... 

For fast reactions Da becomes large. Based on that assumption and standard correlations for mass transfer inside the micro channels, both the model for the micro-channel reactor and the model for the fixed bed can be reformulated in terms of pseudo-homogeneous reaction kinetics. Finally, the concentration profile along the axial direction can be obtained as the solution of an ordinary differential equation. 

This is a homogeneous linear differential equation of second order and its characteristic equation is... 

The application of the time-independent Schrodinger equation to a system of chemical interest requires the solution of a linear second-order homogeneous differential equation of the general form... 

One of die most important second-order, homogeneous differential equations is that of Hennite It arises in the quantum mechanical treatment of the harmonic oscillator. Schrfidinger s equation for the harmonic oscillator leads to the differential equation... 

Example The rate of the homogeneous bimolecular reaction A + B C is characterized by the differential equation dxJdt = k(a — x) b — x), where a = initial... 

The reactor model adopted for describing the lab-scale experimental setup is an isothermal homogeneous plug-flow model. It is composed of 2NP + 2 ordinary differential equations of the type of Equation 16.11 with the initial condition of Equation 16.12, NP + 3 algebraic equations of the type of Equation 16.13, and the catalytic sites balance (Equation 16.14) ... 

An ordinary differential equation has only one variable. Those with more variables are partial differential equations. In most applications to be considered here the differential equations are of the homogeneous type. This means that if yi(x) and 92(2) are two solutions of the equation... 

It is often possible to find a solution of homogeneous differential equations in the form of a power series. According to Frobenius, the power series should have the general form... 

The Thomas-Fermi (TF) model (1927) for a homogeneous electron gas provides the underpinnings of modern DFT. In the following discussion, it will be shown that the model generates several useful concepts, relates the electron density to the potential, and gives a universal differential equation for the direct calculation of electron density. The two main assumptions of the TF model are as follows ... 

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