Differential equations, first-order

Written in matrix notation, the system of first-order differential equations, (A3.4.139) takes the fomi... 

Equation (B2.4.13) is a pair of first-order differential equations, so its fonnal solution is given by equation (B2.4.14)), in which exp() means the exponential of a matrix. 

Here, dj = cos(y,j) and sy = sin(Yy). The three angles are obtained by solving the following three coupled first-order differential equations, which follow from Eq. (19) [36,84,85] ... 

Equation (26) is a set of partial first-order differential equations. Each component of the Curl forms an equation and this equation may or may not be coupled to the other equations. In general, the number of equations is equal to the number of components of the Curl equations. At this stage, to solve this set of equation in its most general case seems to be a fomiidable task. 

In Section V.B, we discussed to some extent the 3x3 adiabatic-to-diabatic transformation matrix A(= for a tri-state system. This matrix was expressed in terms of three (Euler-type) angles Y,y,r = 1,2,3 [see Eq. (81)], which fulfill a set of three coupled, first-order, differential equations [see Eq. (82)]. 

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... 

The net result is that we now have two first-order differential equations of the eigenvalue form ... 

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... 

Quasilinear first-order differential equations are like... 

As an alternative to deriving Eq. (8-2) from a dynamic mass balance, one could simply postulate a first-order differential equation to be valid (empirical modeling). Then it would be necessary to estimate values for T and K so that the postulated model described the reactor s dynamic response. The advantage of the physical model over the empirical model is that the physical model gives insight into how reactor parameters affec t the v ues of T, and which in turn affects the dynamic response of the reac tor. 

Equation (7) is a second-order differential equation. A more general formulation of Newton s equation of motion is given in terms of the system s Hamiltonian, FI [Eq. (1)]. Put in these terms, the classical equation of motion is written as a pair of coupled first-order differential equations ... 

By substimting the definition of H [Eq. (1)] into Eq. (8), we regain Eq. (6). The first first-order differential equation in Eq. (8) becomes the standard definition of momentum, i.e.. Pi = miFi = niiVi, while the second turns into Eq. (6). A set of two first-order differential equations is often easier to solve than a single second-order differential equation. 

In general, consider a system whose output is x t), whose input is y t) and contains constant coefficients of values a, h, c,..., z. If the dynamics of the system produce a first-order differential equation, it would be represented as... 

Equation (2.32) can be expressed as a first-order differential equation... 

From equations (8.15) and (8.16) the set of first-order differential equations are... 

Equations (8.20) and (8.21) are both first-order differential equations, and ean be written in the form... 

Equations 5-64, 5-65, and 5-66 are first order differential equations, whieh require initial or boundary eonditions. Eor the bateh reaetor, these are the initial eoneentrations of A, B, and C. In addition to the initial eoneentrations, the rate eonstants kj and kj are also required to simulate tlieir eoneentrations. The eoneentration profiles depend on die values of kj and kj (i.e, kj = kj, kj > kj, kj > kj). Assume dial at die beginning of die bateh proeess, at time t = 0, =1-0 mol/m, and... 

Equations 5-81, 5-82 and 5-83 are first order differential equations that ean be solved simultaneously using the Runge-Kutta fourth order method. Consider two eases ... 

Equations 5-88, 5-89, and 5-90 are first order differential equations and the Runge-Kutta fourth order method with the boundary eonditions is used to determine the eoneentrations versus time of the eomponents. 

Equations 5-103, 5-104, 5-105, and 5-106 are first order differential equations. The eomputer program BATCH56 simulates the above differential equations with time inerement = At = 0.2 hr. Table 5-5 gives the results of the simulation and Eigure 5-15 shows plots of the eoneentrations versus time. 

Equations 5-110, 5-112, 5-113, and 5-114 are first order differential equations and the Runge-Kutta fourth order numerical method is used to determine the concentrations of A, B, C, and D, with time, with a time increment h = At = 0.5 min for a period of 10 minutes. The computer program BATCH57 determines the concentration profiles at an interval of 0.5 min for 10 minutes. Table 5-6 gives the results of the computer program and Figure 5-16 shows the concentration profiles of A, B, C, and D from the start of the batch reaction to the final time of 10 minutes. 

Equations 5-146, 5-149, and 5-152 are first order differential equations. The eoneentration profiles of A, B, C, and the volume V of the bateh using Equation 5-137 is simulated with respeet to time using the Runge-Kutta fourth order numerieal method. 

Equation 5-336 is a first order differential equation. The behavior of eomponent A ean be predieted with time from the boundary eon-ditions, the flowrate of the feed, eomposition, and the volume of the reaetor. 

Both Equations 5-336 and 5-340 are first order differential equations, whieh ean be solved to determine the transient eoneentration of the anhydride. 

Equation 5-372 is a first order differential equation of the form... 

Equations 6-66 and 6-67, respectively, are two coupled first order differential equations. This is because dC /dt is a function of T and while dT/dt is also a function of and T. The Runge-Kutta fourth... 

Equation 6-74 is a first order differential equation substituting Equations 6-70 and 6-78 for the temperature, it is possible to simulate the temperature and time for various eonversions at AX = 0.05. Table 6-4 gives the eomputer results of the program BATCH63, and Eig-ure 6-7 shows profiles of both fraetional eonversion and temperature against time. The results show that for the endodiermie reaetion of (-i-AH[ /a) = 15.0 keal/gmol, die reaetor temperature deereases as eonversion inereases with time. 

Equations 6-94 and 6-97 are first order differential equations, and it is possible to solve for both the eonversion and temperature of hydrogenation of nitrobenzene relative to the reaetor length of 25 em. A eomputer program PLUG61 has been developed employing the Runge-Kutta fourth order method to determine the temperature and eonversion using a eatalyst bed step size of 0.5 em. Table 6-6 shows... 

Equation 8-90 is a first order differential equation, whieh is of the form dY/dx -i- P(x)y = Q(x). The integrating faetor is In Equation 8-90, the integrating faetor is... 

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