Ordinary Differential Equations of the First Order

Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M(x, y) dx + N(x. y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = j fix) dx + c, which may or may not be expressible in simpler 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 

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This is already an ordinary differential equation of the first order (/(x) is a known deterministic function), but nobody knows how to find G(x, t). 

Table 1.2. Solutions of Ordinary Differential Equations of the First Order... 

As it was mentioned in Section 2.1.1, the concentration oscillations could be simulated quite well by a set of even two ordinary differential equations of the first order but paying the price of giving up the rigid condition imposed on interpretation of mechanisms of chemical reactions namely that they are based on mono- and bimolecular stages only (remember the Hanusse theorem [19]) An example of what Smoes [7] called the heuristic-topological model is the well-known Brusselator [2], Its scheme was discussed in Section 2.1.1 see equations (2.1.33) to (2.1.35). 

The problem of solving the system of 2f ordinary differential equations of the first order, i.e. the canonical equations, is therefore equivalent to that of finding a complete solution of the partial differential equation (3) (/ being greater than 1). This is a special case of general theorems on the relation between ordinary and partial differential equations. 

This chapter deals with the numerical integration of systems of ordinary differential equations of the first order and in the explicit form ... 

Process Transfer Function Models In continuous time, the dynamic behaviour of an ideal continuous flow stirred-tank reactor can be modelled (after linearization of any nonlinear kinetic expressions about a steady-state) by a first order ordinary differential equation of the form... 

If we divide the airshed into L cells and consider N species, LN ordinary differential equations of the form (15) constitute the airshed model. As might be expected, this model bears a direct relation to the partial differential equations of conservation (7). If we allow the cell size to become small, it can be shown that (15) is the same as the first-order spatial finite difference representation of (7) in which turbulent diffusive transport is neglected—i.e,. 

First order series/parallel chemical reactions and process control models are usually represented by a linear system of coupled ordinary differential equations (ODEs). Single first order equations can be integrated by classical methods (Rice and Do, 1995). However, solving more than two coupled ODEs by hand is difficult and often involves tedious algebra. In this chapter, we describe how one can arrive at the analytical solution for linear first order ODEs using Maple, the matrix exponential, and Laplace transformations. 

A transformation of the dependent variables Cjt, and Cs allowed DelBorghi, Dunn, and Bischoff [9] and E>udukovic [25] to reduce the coupled set of partial differential equations for reactions first-order in the fluid concentration and with constant porosity and diffusivity, into a single partial differential equation. With the pseudo-steady-state approximation, this latter equation is further reduced to an ordinary differential equation of the form considered in Chapter 3 on diffusion and reaction (sk Problem 4.2). An extensive collection of solutions of such equations has been presented by Aris [7]. 

However, such an equality is impossible, since by changing one of arguments, for example, R, the first term varies while the second one remains the same, and correspondingly the sum of these terms cannot be equal to zero for arbitrary values of R and 0. Therefore, we have to conclude that neither term depends on the coordinates and each is constant. This fact constitutes the key point of the method of separation of variables, allowing us to describe the function C/ as a product of two functions, each of them depending on one coordinate only. For convenience, let us represent this constant in the form +m, where m is called a constant of separation. Thus, instead of Laplace s equation we have two ordinary differential equations of second order ... 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

This results In a set of first-order ordinary differential equations for the dynamics of the moments. However, the population balance Is still required In the model to determine the three Integrals and no state space representation can be formed. Only for simple MSMPR (Mixed Suspension Mixed Product Removal) crystallizers with simple crystal growth behaviour, the population balance Is redundant In the model. For MSMPR crystallizers, Q =0 and hp L)=l, thus ... 

In simplifying the packed bed reactor model, it is advantageous for control system design if the equations can be reduced to lit into the framework of modern multivariable control theory, which usually requires a model expressed as a set of linear first-order ordinary differential equations in the so-called state-space form ... 

To the best of our knowledge, the first paper devoted to symmetry reduction of the 57/(2) Yang-Mills equations in Minkowski space has been published by Fushchych and Shtelen [27] (see also Ref. 21). They use two conformally invariant ansatzes in order to perform reduction of Eqs. (1) to systems of ordinary differential equations. Integrating the latter yields several exact solutions of Yang-Mills equations (1). 

We will consider a dispersed plug-flow reactor in which a homogeneous irreversible first order reaction takes place, the rate equation being 2ft = k, C. The reaction is assumed to be confined to the reaction vessel itself, i.e. it does not occur in the feed and outlet pipes. The temperature, pressure and density of the reaction mixture will be considered uniform throughout. We will also assume that the flow is steady and that sufficient time has elapsed for conditions in the reactor to have reached a steady state. This means that in the general equation for the dispersed plug-flow model (equation 2.13) there is no change in concentration with time i.e. dC/dt = 0. The equation then becomes an ordinary rather than a partial differential equation and, for a reaction of the first order ... 

We now have to solve the following system of two nonlinear coupled first-order ordinary differential equations for the given initial conditions ... 

A system of differential and algebraic equations (DAE system) is obtained from the model balances. The developed set of equations consists of the ordinary differential equations of first order and of partial differential equations. An analytical solution of the coupled equations is not possible. Therefore, a numeric procedure is used. 

Many drugs undergo complex in vitro drug degradations and biotransformations in the body (i.e., pharmacokinetics). The approaches to solve the rate equations described so far (i.e., analytical method) cannot handle complex rate processes without some difficulty. The Laplace transform method is a simple method for solving ordinary linear differential equations. Although the Laplace transform method has been used for more complex applications in physics, engineering, and other research areas, here it will be applied to ordinary differential equations of first-order rate processes. 

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