Differential equations with constant coefficients

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),... 

Neither of these equations tells us which spin is on which electron. They merely say that there are two spins and the probability that the 1, 2 spin combination is ot, p is equal to the probability that the 2, 1 spin combination is ot, p. The two linear combinations i i(l,2) v /(2,1) are perfectly legitimate wave functions (sums and differences of solutions of linear differential equations with constant coefficients are also solutions), but neither implies that we know which electron has the label ot or p. 

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. 

The number of independent rate equations is the same as the number of independent stoichiometric relations. In the present example. Reactions (1) and (2) are reversible reactions and are not independent. Accordingly, C,. and C, for example, can be eliminated from the equations for and which then become an integrable system. Usually only systems of linear differential equations with constant coefficients are solvable analytically. 

The unsteady material balances of tracer tests are represented by linear differential equations with constant coefficients that relate an input function Cj t) to a response function of the form... 

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

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 preceding two equations are examples of linear differential equations with constant coefficients and their solutions are often found most simply by the use of Laplace transforms [1]. 

Suppose we have a simple differential equation with constant coefficients... 

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

In this section, we will outline only those properties of the Laplace transform that are directly relevant to the solution of systems of linear differential equations with constant coefficients. A more extensive coverage can be found, for example, in the text book by Franklin [6]. 

Compartmental analysis is the most widely used method of analysis for systems that can be modeled by means of linear differential equations with constant coefficients. The assumption of linearity can be tested in pharmaeokinetic studies, for example by comparing the plasma concentration curves obtained at different dose levels. If the curves are found to be reasonably parallel, then the assumption of linearity holds over the dose range that has been studied. The advantage of linear... 

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... 

What are some of the mathematical tools that we use In classical control, our analysis is based on linear ordinary differential equations with constant coefficients—what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. 

This pair of differential equations may be solved by differentiating the first with respect to time and eliminating Aft and J(Aft )/Jr between the derived equation and the two original equations in order to arrive at a second-order differential equation with constant coefficients. 

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

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

Therefore the last term in Eq. (6.37) is equal to zero. We end up with a linear ordinary differential equation with constant coefficients in terms of perturbation variables. 

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

Exercise. Show that dtP x, t) = a does not tend to a stationary solution when a > 0 and does not conserve positivity when a < 0. Extend this conclusion to general differential equations with constant coefficients. 

Find the general solutions to linear second-order differential equations with constant coefficients by substitution of trial functions... 

Second-order Differential Equations with Constant Coefficients... 

The equilibrium equations of a hyperboloid of revolution used for cooling towers derived by using membrane theory under an arbitrary static normal load are reduced to a single partial differential equation with constant coefficients. The problem of finding displacements is reduced to a similar type of equation so that the solution for this problem becomes straightforward. 11 refs, cited. 

The calculating procedure is based on sub-division of the Arctic Basin into grids (Eijk. This is realized by means of a quasi-linearization method (Nitu et al., 2000a). All differential equations of the SSMAE are substituted in each box E by easily integrable ordinary differential equations with constant coefficients. Water motion and turbulent mixing are realized in conformity with current velocity fields which are defined on the same coordinate grid as the E (Krapivin et al., 1998). 

Equation 1.12 is a system of linear differential equations with constant coefficients. Then, following the rules for solving this type of an equation, its solution can be written in the following form [7] ... 

L. Onsager was the first person to formulate in 1931 the principle of interacting thermodynamic processes. The underlying idea is that the rate of numerous interacting irreversible processes can be described by linear differential equations with constant coefficients ... 

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