Differential equations linear, order

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

No assumption has been made as to continuity, in general, but it will now be assumed that all functions have continuous derivatives of order n + 1. Then the t satisfy a linear ordinary differential equation of order n + 1, which can be written in the form... 

If the dependent variable y(jt) and all of its derivatives occur in the first degree and do not appear as products, the equation is said to be linear. In effect, the solution of a differential equation of order n necessitates n integrations, each of which involves an arbitrary constant. However, in some cases one or more of these constants may be assigned specific values. The results, which are also solutions of the differential equation, are referred to as particular solutions. The general solution, however, includes all of the n constants of integration, whose evaluation requires additional information associated with the application. 

Use the integrating factor method to find the general solutions to first-order differential equations linear in y... 

This is a linear differential equation of order n and has a general solution obtained by setting the right-hand side equal to zero. The equation is solved in Sec. III.6. To this general solution we must add a particular solution. Since Mi is a known function of z [Eq. (III.7B.7) or (III.7B.9)], the particular solution (z) can be found by known methods. In this case it has the form... 

In order to understand the mathematical importance of the chemostat, one must look at the broader picture of the subject of nonlinear differential equations. Linear differential equations have been studied for more than two hundred years their solutions have a rich structure that has been well worked out and exploited in physics, chemistry, and biology. Avast and challenging new world opens up when one turns to nonlinear differential equations. There is an almost incomprehensible variety of non-linearities to be studied, and there is little common structure among them. Models of the physical and biological world provide classes of nonlinearities that are worthy of study. Some of the classic and most studied nonlinear differential equations are those associated with the simple pendulum. Other famous equations include those associated with the names of... 

A linear differential equation, however, may contain (1) powers or nonlinear functions of the independent variable, such as a and cos x and (2) products of the dependent variable (or its derivatives) and functions of the independent variable, such as r y, x, and A linear differential equation of order n can be expressed in the most general form as... 

The solution to this example satisfies the differential equation no matter what values Cl and C2 have. It is actually a family of functions, one function for each set of values for ci and C2. A solution to a linear differential equation of order n that contains n arbitrary constants is known to be a general solution. A general solution is a family of functions which includes almost every solution to the differential equation. The solution of Eq. (8.20) is a general solution, since it contains two arbitrary constants. There is only one general solution to a differential equation. If you find two general solutions for the same differential equation that appear to be different, there must be some mathematical manipulations that will reduce both to the same form. A solution to a differential equation that contains no arbitrary constants is called a particular solution. A particular solution is usually one of the members of the general solution, but it might possibly be another function. 

Equation 4.5-4 is now inserted into the equation for the concentration of A inside the pore, Eq. 4.5-1, leading to a second-order differential equation linear in Ca, but containing Riy) in the group multiplying Ca,- With R(y) = Rp at t = 0, the equation can be solved for the initial concentration profile of. 4 in the pore. 

In Problem 1 the reader was called upon to integrate simple first order and second order rate laws. In this section we will develop techniques for dealing with more complicated differential equations linear and nonlinear, with two or more dependent variables. 

The method of determining T via amplitude modulation of Hi relies on variation of the modulation frequency, denoted by QJIti, until it exceeds T), at which point the magnetization cannot respond to the power variation and there is a loss in the detected EPR signal amplitude (Herve Pescia, 1960a). In a sense, this Ti measurement is analogous to that used to analyze the impedance of a nonlinear system, sueh as a passive filter. The precept is that the response of a system y t) to some perturbation x t) is determined by some differential equation of order n. In the case of a linear system and perturbation x(t)=A sin((oO, one observes a response y(t)=B sin(co -l-(t)) and one defines a transfer function as the ratio of output to input (in the frequency domain) H(j( i)= H(( i) wherey((o) and x(co)... 

In general, when a linear differential equation of order p is converted to discrete time, a linear difference equation of order p results. For example, consider the second-order model ... 

In another research, the thermo-mechanical behavior of SMPs was described by both linear and nonlinear viscoelastic theories [4]. In this woik four element mechanical units consisting of spring, dashpot and frictional device were used to derive a constitutive differential equation. In order to determine the material properties by a constitutive differential equation the modulus, viscosity and other parameters were assumed to decay exponentially with temperature. Liu et al. [5] developed a constitutive equation for SMPs based on thermodynamic concepts of entropy and internal energy. They adopted the concept of frozen strain and demonstrated the utility of the model by simulating the stress and strain... 

Ordinary differential equations may be categorized as linear and nonlinear equations. A differential equation is nonlinear if it contains products of the dependent variable or its derivatives or of both. For example, Eqs. (5.21) and (5,22) are nonlinear because they contain the terms y dyldx) and (dyldxf, respectively, whereas Eq. (5.20) is linear. The general form of a linear differential equation of order n may be written as... 

On subsciCuLlng (12.49) into uhe dynamical equations we may expand each term in powers of the perturbations and retain only terms of the zeroth and first orders. The terms of order zero can then be eliminated by subtracting the steady state equations, and what remains is a set of linear partial differential equations in the perturbations. Thus equations (12.46) and (12.47) yield the following pair of linearized perturbation equations... 

Solve the following seeond order linear differential equation subjeet to the speeified "boundary eonditions" ... 

The solution of equation 16 is a decreasing, simple exponential where = k ([A ] + [P ]) + k. The perturbation approach generates small deviations in concentrations that permit use of the linearized differential equation and is another instance of pseudo-first-order behavior. Measurements over a range of [A ] + [T ] allow the kineticist to plot against that quantity and determine / ftom the slope and from the intercept. 

Linear Equations A differential equation is said to be linear when it is of first degree in the dependent variable and its derivatives. The general linear first-order differential equation has the form dy/dx + P x)y = Q x). Its general solution is... 

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... 

FIG. 7-1 Constants of the power law and Arrhenius equations hy linearization (a) integrated equation, (h) integrated fimt order, (c) differential equation, (d) half-time method, (e) Arrhenius equation, (f) variahle aotivation energy, and (g) ehange of meohanism with temperature (T in K),... 

Although the differential equation is first-order linear, its integration requires evaluation of an infinite series of integrals of increasing difficulty. 

Nonlinear versus Linear Models If F, and k are constant, then Eq. (8-1) is an example of a linear differential equation model. In a linear equation, the output and input variables and their derivatives only appear to the first power. If the rate of reac tion were second order, then the resiilting dynamic mass balance woiild be ... 

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. 

Like thermal systems, it is eonvenient to eonsider fluid systems as being analogous to eleetrieal systems. There is one important differenee however, and this is that the relationship between pressure and flow-rate for a liquid under turbulent flow eondi-tions is nonlinear. In order to represent sueh systems using linear differential equations it beeomes neeessary to linearize the system equations. 

Therefore, the slope of the linear plot Cg versus gives the ratio kj/kj. Knowing kj -i- kj and kj/kj, the values of kj and kj ean be determined as shown in Figure 3-10. Coneentration profiles of eom-ponents A, B, and C in a bateh system using the differential Equations 3-95, 3-96, 3-97 and the Runge-Kutta fourth order numerieal method for the ease when Cgg =Cco = 0 nd kj > kj are reviewed in Chapter 5. 

Equation 3-133 is a first order linear differential equation of the form dy/dx -i- Py = Q. The integrating factor is IF = and... 

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... 

Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

A water body is considered to be a one-diiuensional estuary when it is subjected to tidal reversals (i.e., reversals in direction of tlie water quality parameter are dominant). Since the describing (differential) equations for the distribution of eitlier reactive or conserv ative (nomciictive) pollutants are linear, second-order equations, tlie principle of superposition discussed previously also applies to estuaries. The principal additional parameter introduced in the describing equation is a tid il dispersion coefficient E. Methods for estimating this tidiil coefficient are provided by Thomaim and Mueller... 

Equation (8.4.3) is a linear first-order differential equation of concentration and reactor length. Using the separation of variables technique to integrate (8.4.3) yields... 

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. 

The first term on the right-hand side of this equation is zero, since it is simply the sum of the electrical charge in solution, which must be zero for a neutral electrolyte solution. The third term is also zero for electrolytes with equal numbers of positive and negative ions, such as NaCl and MgSC>4. It would not be zero for asymmetric electrolytes such as CaCE. However, in the Debye-Huckel approach, all terms except the second are ignored for all ionic solutions. Substitution of the resulting expression into equation (7.20) gives the linear second-order differential equation... 

The change of n, with time was calculated according to first-order kinetics. It is given by a system of r linear differential equations and 0 r(r - 1) variables ... 

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