Simultaneous equations with constant coefficients

By way of praotioe it will be convenient to study a few more examples of simultaneous equations, since they are so common in many branohes of physics. The motion of a partiole in space is determined by a set of three differential equations which determine the position of the moving particle at any instant of time. Thus, if X, Y, Z, represent the three components of a force, F, aoting on a particle of mass m, Newton s law, page 396, tells us that 

In order to solve a set of simultaneous equations, there must be the same number of equations as there are independent variables. Quite an analogous thing occurs with the simultaneous equations in ordinary algebra. The methods used for the solution of these equations are analogous to those employed for similar equations in algebra. The operations here involved are chiefly processes of elimination and substitution, supplemented by differentiation or integration at various stages of the computation. The use of the symbol of operation D often shortens the work. 

The result represents the motion of a particle in an elliptic path, subject to a central gravitational force. 

Divide the one equation by the other expand e Kx reject all but the first three terms of the series. 

from page 405, and remembering that x = 0, when t = 0, 

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Systems of reversible first-order reactions lead to sets of simultaneous linear differential equations with constant coefficients. A solution may be obtained by means of a matrix formulation that is widely used in quantum mechanics and vibrational... 

The linearisation of the non-linear component and energy balance equations, based on the use of Taylor s expansion theorem, leads to two, simultaneous, first-order, linear differential equations with constant coefficients of the form... 

G. Simultaneous linear differential equations with constant coefficients. Take the pair... 

These simultaneous first-order differential equations can be written as a single second-order differential equation with constant coefficients,... 

In the course of developing models for the impedance response of physical systems, differential equations are commonly encountered that have complex variables. For equations with constant coefficients, solutions may be obtained using the methods described in the previous sections. For equations with variable coefficients, a numerical solution may be required. The method for numerical solution is to separate the equations into real and imaginary parts and to solve them simultaneously. 

Equation (2.3) describes a set of linear homogeneous simultaneous differential equations with constant coefficients, which is readily solved once the eigenvalues and eigenvectors of Q are known. Let the matrix Q be diagonalised by the transformation... 

The constants Aj and A2 are known as Lagrange multipliers. As we have already seen two of the variables can be expressed as functions of the third variable hence, for example, dxx and dx2 can be expressed in terms of dx3, which is arbitrary. Thus Ax and A2 may be chosen so as to cause the vanishing of the coefficients of dxx and dx2 (their values are obtained by solving the two simultaneous equations). Then since dx3 is arbitrary, its coefficient must vanish in order that the entire expression shall vanish. This gives three equations that, together with the two constraint equations gt = 0 ( = 1,2), can be used to determine the five unknowns xx, x2, Xg, Xx, and A2. 

Thus, if the saturated vapor pressure is known at the azeotropic composition, the activity coefficient can be calculated. If the composition of the azeotrope is known, then the compositions and activity of the coefficients at the azeotrope can be substituted into the Wilson equation to determine the interaction parameters. For the 2-propanol-water system, the azeotropic composition of 2-propanol can be assumed to be at a mole fraction of 0.69 and temperature of 353.4 K at 1 atm. By combining Equation 4.93 with the Wilson equation for a binary system, set up two simultaneous equations and solve Au and A21. Vapor pressure data can be taken from Table 4.11 and the universal gas constant can be taken to be 8.3145 kJ-kmol 1-K 1. Then, using the values of molar volume in Table 4.12, calculate the interaction parameters for the Wilson equation and compare with the values in Table 4.12. 

With the help of operational stability constants valid for seawater (Table 6.7), a direct computation of the major inorganic species (without using individual activity coefficients) was carried out. The calculations consist essentially in solving 20 equations with 20 unknowns simultaneously. Twelve stability constants are available and nine mass balance relations (of the type... 

Steady state heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear elliptic partial differential equation. For linear parabolic partial differential equations, finite differences can be used to convert to any given partial differential equation to system of linear first order ordinary differential equations in time. In chapter 5.1, we showed how an exponential matrix method [3] [4] [5] could be used to integrate these simultaneous equations... 

In order to investigate the dependence of a on dissolved CO2 and pH, we prepared solutions of simpler composition in which the concentrations of several important free ions and ion-pairs were maximized. In order to do this, we calculated the chemical speciation of carbon-ate/bicarbonate in seawater and in solutions of KCl, NaCl, NgCl2 end mixtures of these salts as a function of pH in the range 6.0 to 8.9. Ion strengths were kept equivalent to those of natural seawater (y 0.7 at S = 35 / <). For the seawater calculations, we assumed the concentrations of major cations given by Culkin (5) and a total CO2 concentration of 2.333 x 10 N. We used an iterative con uter program to solve a series of simultaneous equations based on thermodynamic association constants and estimated activity coefficients at ionic strengths of solutions for free carbonate and bicarbonate, and for the carbonate/bicarbonate ion-pairs with Mg, Ca and Na (6). Our calculations did not consider the reduction in available cations... 

Note that if the concentrations and reaction rates have already been computed and stored, then this is a set of linear differential equations (with non-constant coefficients) for the sensitivity coefficients. Note also that each sensitivity coefficient depends only on the sensitivity coefficients of the other species to the same parameter, but not on sensitivity coefficients with respect to other parameters. Thus, the sensitivity coefficients with respect to a given parameter (K of them) are coupled and must be computed simultaneously, but it is not necessary to solve for all the sensitivity coefficients (2 x / x of them) simultaneously. 

A model relates the output (the dependent variable or variables) to the independent variable(s). Each equation in the model usually includes one or more coefficients that are presumed constant. The term parameter as used here means coefficient and possibly input or initial condition. With the help of experimental data, we can determine the form of the model and subsequently (or simultaneously) estimate the value of some or all of the parameters in the model. 

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