Ordinary differential equations constant coefficients

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

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. 

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. 

For a fixed value of x, this is a third-order, nonlinear, ordinary differential equation. Recall that the metric coefficient h x,y) depends on both coordinates, but that the cone angle 9 is a fixed constant. 

This is a functional equation for the boundary position X and the unknown constant parameter n. Upon substituting Eq. (256) into Eq. (251) an ordinary differential equation is obtained for X(t, n), and a family of curves in the phase plane (X, X) can be obtained. For n sufficiently close to unity two functions in the phase plane can be determined which serve as upper and lower bounds for the trajectories. The choice is guided by reference to the exact solution for the limiting case of constant surface temperature. It is shown that the upper and lower bounds are quite close to the one-parameter phase plane solution, although no comparison is made with a direct numerical solution. The one-parameter solution also agrees well with experiments on the solidification of aluminum under conditions of low surface heat transfer coefficient (hi = 0.02 cm.-1). 

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). 

The governing equations - that is, mainly the component and the total mass balances in the anode channels - are provided here in dimensionless form. The five ordinary differential equations (ODE) with respect to the spatial coordinate describe the development of the five unknowns in one single anode channel, namely the mole fractions, with i = CH4, H2O, H2, CO2, as well as the molar flow density inside the anode channel, y. Here, the Damkohler numbers, Da/, are the dimensionless reaction rate constant of the reforming and the oxidation reaction, respectively, the rj are the corresponding dimensionless reaction rates, and the v, j are the stoichiometric coefficients ... 

More recently, Zou and Zindler (2000) provide a tractable analytic solution to both McKenzie s and Williams and Gill s equations for the hmited problem of constant partition coefficients and porosities. By making all adjustable parameters constant, these models exploit the analytic solutions afforded to linear systems of ordinary differential equations like Equations (l)-(3) (a similar approximation for the transport models described in Section 3.14.4.3.2 can be found in the appendix of Spiegelman and Elliott (1993)). However, because the solutions of these models are sufficiently sensitive to parameters such as DIq it can be challenging to choose specific constants that are actually appropriate to a problem where the properties of the system are expected to change in space (see Section 3.14.4.1). 

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... 

First-Order Linear Ordinary Differential Equation / 2.3.2 Second-Order Linear ODEs with Constant Coefficients / 2.3.3 Nth-Order Linear ODEs with Constant Coefficients... 

These boundary conditions are particularly convenient to evaluate the integration constants, as illustrated below. The mass transfer equation corresponds to a second-order linear ordinary differential equation with constant coefficients. The analytical solution for I a is... 

The mass balance with diffusion and first-order chemical reaction, given by (24-12), is classified as a frequently occurring second-order linear ordinary differential equation (i.e., ODE) with constant coefficients. It is a second-order equation because diffusion is an important mass transfer rate process that is included in the mass balance. It is linear because the kinetic rate law is first-order or pseudo-first-order, and it is ordinary because diffusion is considered only in one coordinate direction—normal to the interface. The coefficients are constant under isothermal conditions because the physicochemical properties of the fluid don t change... 

Differential equations may be either first order, or second order, (third order and others are also possible, but less likely), depending on the derivative levels that appear in the equation. For instance, a differential eqnation commonly appearing in biological systems is the first order constant coefficient linear ordinary differential equation... 

For the phenomena presented in Table 2.1, efficiency can be related to characteristic times by writing a balance of the extensity concerned. For a chemical plug-flow reactor (with an apparent first-order reaction or with heat/mass transfer at constant exchange coefficient), the quantity of this extensity is linearly related to its variation with respect to the reference time, yielding ordinary differential equations such as... 

Observe that Equation (10.37) can be solved for by the method of eigenvectors [2]. It can be seen that nontrivial equations to the set of ordinary differential equations with constant coefficients represented by Equation (10.36) exist only for certain specific values of K called eigenvalues. These are the solutions to... 

Equation (10.50) requires that e 0. Equation (10.49) has the form of the solution to the set of ordinary differential equations with constant coefficients requires that Z 0. It can be seen from Equation (10.50) that ft is an eigenvalue of the rate matrix modified by the free radical concentrations. The free radical concentrations can be obtained from Equation (10.39). The modified rate matrix transposed, is a square matrix of nxn. Therefore, n eigenvalues can be expected. The eigenvalues must obey... 

The equations of motion resulting for the iQ, t)Q Xt)" " Q, t )Y then form a closed set of second-order ordinary differential equations with constant coefficients that are readily solved via Fourier transform techniques. The required for a given molecule will in... 

Equation (7.275) is an ordinary differential equation with constant coefficients. Its solution is easily shown to be... 

Equation 2.20 is a linear ordinary differential equation for (p 9) with constant coefficients whose solution is... 

This is a linear, first-order, ordinary differential equation with constant coefficients and a constant forcing term, for which we can immediately write the solution (after a bit of algebra) as... 

The analysis of many physicochemical systems yields mathematical models that are sets of linear ordinary differential equations with constant coefficients and can be reduced to the form... 

Sets of linear ordinary differential equations with constant coefficients have closed-form solutions that can be readily obtained from the eigenvalues and eigenvectors of the matrix A. 

Tracer response is formulated as an unsteady material balance in terms of linear differential equations with constant coefficients that relate an input function, Cf(t), to a response function, C(t). Such equations of ordinary type have the form... 

Initiated by photorecovered FMNH2, the bioluminescent reaction is a short flash of light with pronounced maximum. In this case the kp value is very large" and the enzyme makes, at this, one cycle. It means, that interaction of enzyme with FMNH2 very quickly decreases. So the kinetic behavior of the system can be described as the system of ordinary linear differential equations with constant coefficients ... 

Certain types of differential equations occur sufficiently often to justify the use of formulas for the corresponding particular solutions. The following set of Tables I to XIV covers all first, second, and th order ordinary linear differential equations with constant coefficients for which the right members are of the form F[x)e sinsx or P x)e cossx, where r and s are constants and P(x) is a polynomial of degree n. 

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