Order of a differential equation

The order of a differential equation is the order of its highest derivatives, which in the example quoted is a second-order derivative, d W/dP. 

A differential equation is ordinary or partial, according as there is one or more than one independent variables present. Ordinary differential equations will be treated first. Equations like (2) and (3) above are said to be of the first order, because the highest derivative present is of the first order. For a similar reason (4) and (6) are of the second order, (5) of the third order. The order of a differential equation, therefore, is fixed by that of the highest differential coefficient it contains. The degree of a differential equation is the highest power of the highest order of of differential coefficient it contains. This equation is of the second order and first degree ... 

The order of a differential equation corresponds to the highest derivative, whereas the degree is associated with the power to which the highest derivative is raised. 

Whichever the type, a differential equation is said to be of /ith order if it involves derivatives of order n but no higher. The equation in the first example is of first order and that in the second example of second order. The degree of a differential equation is the power to which the derivative of the highest order is raised after the equation has been cleared of fractions and radicals in the dependent variable and its derivatives. 

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. 

Aitemativeiy, the beam end couid have compiete rotational restraint and no transverse displacement, i.e., clamped. However, a third boundary condition exists in Rgure D-3 just as in Figure D-2. That is, an axial condition on displacement or force must exist in addition to the conditions usually thought of as comprising a clamped-end condition. Note that the block-like device at the end of the beam prevents rotation and transverse deflection. A similar device will be used later for plates. Whether all of the three boundary conditions can actually be enforced depends on the order of the differential equation set when (necessarily approximate) force-strain and moment-curvature relations are substituted in Equations (D.2), (D.4), and (D.7). 

A rigorous definition of stability of a difference scheme will be formulated in the next section. The improvement of the approximation order for a difference scheme on a solution of a differential equation will be of great importance since the scientists wish the order to be as high as possible. 

Equation (3) is in the form of a differential equation describing a first-order kinetic process, and, as a result, drug absorption generally adheres to first-order kinetics. The rate of absorption should increase directly with an increase in drug concentration in the GI fluids. 

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. 

As the number of boundary conditions is usually the same as the order of the differential equation for a particular chemical problem, there will be no undetermined constants of integration associated with the solution. 

We allow ourselves a short digression here, in order to make a special point. There are two ways of presenting an error in a numerical solution of a differential equation. The usual way is to refer to the error in the quantity computed at each new time interval that is, the difference between the numerical approximation and the exact solution (if it is known). Another way is to compute, for each calculated value, the time at which that value is exact, and to express the error as a time shift, the difference between the calculated time and the exact time at that iteration number. It is called a time shift because in many kinds of simulations dealt with in this book, time itself does not enter the equations and, once a simulated sequence of values has become shifted along in time, that shift is permanent. Putting this another way, there is no clock inherent in the method. It will be seen (Chap. 8) that in fact, in... 

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 differential equation entered can be of any order. For a differential equation of order N, N boundary conditions have to be specified. The numerical solution can be stored in a variable and can be used later for plotting purposes as shown in the following examples. 

The Least squares method is generally applied to problems containing first order derivatives. A differential equation containing second order derivatives is thus re-written as a set of two equations containing only first order derivatives. 

For example, the free particle wavefunction (2.200), a solution of a differential equation of the second-order, is also characterized by two coefficients, and we may choose 5 = 0 to describe a particle going in the positive x direction or ff = 0 to describe a particle going in the opposite direction. The other coefficient can be chosen to express normalization as was done in Eq. (2.82). 

Generally, T(x,t) satisfies a nonlinear bonndary-valne problem. We obtain a singular perturbation problem when, npon letting s 0, the order of the differential equation is reduced. 

The quantity of dine / dt represents an activation energy barrier against the flow. A spectrum of relaxation times increases the order of the differential equation to n, where n is the number of relaxation times. The equation of kinetics is no longer exponential and all interpretations become difficult, for example... 

It is not difficult to show that the complete integral of a differential equation of the nth order, contains n, and no more than n, arbitrary constants. As the reader acquires experience in the representation of natural processes by means of differential equations, he will find that the integration must provide a sufficient number of undetermined constants to define the initial conditions of the natural process symbolized by the differential equation. The complete solution must provide so many particular solutions (containing no undetermined constants) as there are definite conditions involved in the problem. For instance, equation (5), page 375, is of the third order, and the complete solution, equation (9), requires three initial conditions, g, s0, v0 to be determined. Similarly, the solution of equation (4), page 375, requires two initial conditions, m and 6, in order to fix the line. 

The general solution of a differential equation of nth order usually has n arbitrary constants. To fix these constants, we may have boundary conditions, which are conditions that specify the value of y or various of its derivatives at a point or points. Thus, if y represents the displacement of a vibrating string held fixed at two points, we knowy must be zero at these points. 

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