Second-order linear with constant

Eq (9) is non-homogeneous second order linear with constant coefficients and can be solved readily. Boundary conditions are,... 

Equation 9.18 is a linear, second-order ODE with constant coefficients. An analytical solution is possible when the reaction is first order. The general solution to Equation 9.18 with 3 A = —ka is... 

Eq. (11.36) can be transformed into a linear second-order equation with constant coefficients... 

We will consider here linear second-order equations with constant coefficients, in which the functions p(x) and q x) in Eq. (8.2) are constants. The more general case gives rise to special functions, several of which we will encounter... 

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. 

The ODEs are linear with constant coefficients. They can be converted to a single, second order ODE, much like Equation (11.22), if an analytical solution is desired. A numerical solution is easier and better illustrates what is necessary for anything but the simplest problem. Convert the independent variable to dimensionless form, = z/L. Then... 

Second-Order Linear ODEs With Constant Coefficients... 

The [Ruv(N40)(0)]2+ complex is shown to oxidize a variety of organic substrates such as alcohols, alkenes, THF, and saturated hydrocarbons, which follows a second-order kinetics with rate = MRu(V)][substrate] (142). The oxidation reaction is accompanied by a concomitant reduction of [Ruv(N40)(0)]2+ to [RuIII(N40)(0H2)]2+. The mechanism of C—H bond oxidation by this Ru(V) complex has also been investigated. The C—H bond kinetic isotope effects for the oxidation of cyclohexane, tetrahydrofuran, propan-2-ol, and benzyl alcohol are 5.3 0.6, 6.0 0.7, 5.3 0.5, and 5.9 0.5, respectively. A mechanism involving a linear [Ru=0"H"-R] transition state has been suggested for the oxidation of C—H bonds. Since a linear free-energy relationship between log(rate constant) and the ionization potential of alcohols is observed, facilitation by charge transfer from the C—H bond to the Ru=0 moiety is suggested for the oxidation. 

The THG experiments of the pure PTA (TMS) series show a power law increase of the second-order hyperpolarizability with an exponent a=2.52 0.10 for the oligomers n=1 to 6. The polydisperse PTA polymer samples exhibit a constant second-order hyperpolarizability per monomer unit of y/ =4.1xl0 48 m5/V2 (300x10 36 esu) indicating an only linear increase for longer PTAs. The... 

The selective formation of F1H2 + is well characterized by the ESR spectra [84], No reaction occurs without acid [84]. Since FI is already protonated at the oxidized state in the presence of HCIO4 in MeCN, the rate constant of electron transfer (/cet) from cw-[R2Co(bpy)2]+ to F1H+ increases linearly with [H+] [84], in contrast to the case of the acid-catalyzed reduction of Q, in which the kgt value shows the second-order dependence with respect to [H+] when QH is further protonated to give QH2-+ [77],... 

The physical argument presented above is consistent with the mathematical nature of the problem since tlie heat conduction equation is second order (i.e., involves second derivative.s with respect to the space variables) in all directions along which heat conduction is significant, and the general solution of a second-order linear differential equation involves two surbitrary constants for each direction. That is, the number of boundary conditions that needs to be specified in a direction is equal to the order of the differential equation in that direction. 

The adopted heat capacity values are based on the studies by Montgomery (1, 405.79-433.31 K) and West (2, 373-678 K). Liquid sulfur undergoes a second order transition with a maximum reported at 432.02 0.20 K (1 ) and 432.25 0.30 K (2) this has been attributed to the polymerization of Sg molecules (3). We adopt the tabulated heat capacity values of Montgomery ( ) up to 434 K and those of West (2) above 434 K. The heat capacity is assumed to be constant at 7.568 cal mol above 810 K. Below T 388.36 K, the heat capacity values are obtained by linear extrapolation using the slope of the values in the region T to 420 K. The entropy is calculated in a manner similar to that used for the enthalpy of formation. 

The corresponding rate constants are plotted as a function of ethylene concentration in Figure 172. The initiation constants (fc2 and k2) become linear with ethylene concentration when the square root is taken, indicating that the dependence on ethylene concentration of the initiation reactions is approximately second-order. The decay constant (fc3) was unchanged with ethylene concentration, indicating a zero-order dependence, and the polymerization itself (R) was directly proportional to the ethylene concentration, indicating a first-order dependence. 

The mass balance for a first-order reaction in a tubular reactor with a flow velocity of v and the concentration of reactant in feed, Cp undergoing a reaction with the rate constant of k is governed by the following second-order linear equation with constant coefficients ... 

The reactivity of the model phenols and benzyl alcohols with phenyl isocyanate was determined in the presence of a tertiary amine (DMCHA) and a tin catalyst (DBTDL) by measurement of the reaction kinetics. The experimental results based on initial equal concentrations of phenyl isocyanate and protic reactants showed that the catalyzed reactions followed second order reaction with respect to the disappearance of isocyanate groups (see Figure 1). It was also found that a linear relationship exists between the experimental rate constant kexp, and the initial concentration of the amine catalyst (see Figure 2). In the case of the tin catalyst, the reaction with respect to catalyst concentration was found to be one-half order (see Figures 3-4). A similar relationship for the tin catalyzed urethane reaction was found by Borkent... 

In Table 9.4 the slopes of the linear part of the reaction profiles (u), the second-order kinetic rate constants (k) and the initial rate constants (/c0) are summarized. From Table 9.4 it is clear that for the reactions of NBDPZ with monoisocyanates 11 and 12 and diisocyanate 13 the rate constants calculated by fitting the data to second-order kinetics are in good agreement with those calculated based on initial rates only, supporting the second-order kinetics of the reactions. The experimental data were also fitted to a first-order kinetics model. From the fits, which are not shown here, and the residual values, it was clear that only the second-order reactions kinetics could describe the data well. 

If the functions and (or alternatively the whole set of unperturbed functions are expressed in terms of configurations by the Cl method, it is seen that a second order quantity should be more sensitive to correlation than a one-electron expectation value, because the matrix element of contains the coefficients of the Cl expansion for and linearly. It may happen that the quantity to be calculated (e.g. a magnetic susceptibility or a magnetic shielding constant) consists of two parts, a second-order one with respect to an external perturbation and a first-order one with respect to another operator... 

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... 

Oxidation of [Ru302(NH3)i4] or [Ru302(NH3),o(en)2f with HCl in air, or by CI2 water gives the [4,3,4] " analogue ruthenium brown. Mossbauer studies on the 6-H and 7-1- ions are in agreement with linear structures for these complexes. The resonance Raman spectra of ruthenium red and brown and of their en analogues have been reported. Ruthenium brown is reduced by OH in a second order reaction with rate constant k = 1.9 x 10 s mol , = 79.5kJmol ,... 

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... 

So far we have considered only cases where the potential energy V(ac) is a constant. Hiis makes the SchrSdinger equation a second-order linear homogeneous differential equation with constant coefficients, which we know how to solve. However, we want to deal with cases in which V varies with x. A useful approach here is to try a power-series solution of the SchrSdinger equation. 

In the limit as Ax approaches zero, a second-order linear homogeneous equation with constant coefficients is obtained ... 

A del is not present in all quality measurements. Consider, for example, a pH measurement. The response of a pH electrode to a change in pH can usually be characterized by a second-order response with a small time constant of the order of a few seconds and a larger time constant of the order of 5-30 seconds, depending on the type of electrode. However, the most important characteristic of the electrode is that it gives a non-linear response the time constants for a positive change in pH and negative change in pH can be considerably different. 

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