General solution to a differential equation

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. 

We are not finished with a problem when we find a general solution to a differential equation. We usually have additional information that will enable us to pick a particular solution out of the family of solutions. Such information consists of knowledge of boundary conditions and initial conditions. Boundary conditions arise from physical requirements on the solution, such as necessary conditions that apply to the boundaries of the region in space where the solution applies, or the requirement that the value of a physically measurable quantity must be a real number. Initial conditions arise from knowledge of the state of the system at some initial time. 

In this case, Oq is the maximum amplitude of the stress. The solution to this differential equation will give a functional description of the strain in this dynamic experiment. In the following example, we examine the general solution to this differential equation. 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Hence the general solution to the differential equation for Vp in the neighbourhood of a critical point is as given in eqn (E2.14). 

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. 

Because Equation (3.48) is a quadratic, there are two independent solutions to the auxiliary equation. Thus, the most general solution to the differential equation given by Equation (3.46) is a linear combination, as shown in Equation (3.49), where a and b are weighting constants. 

This is a differential equation. It is called linear because the dependent variable x enters only to the first power and is called second order because its highest-order derivative is the second derivative. The solution to a differential equation is a function that gives the dependent variable (x in this case) as a function of the independent variable (t in this case). There can be more than one solution function for a given differential equation. The general solution of a differential equation is a family of functions that includes nearly every solution of the equation. 

A final type of problem to be considered in this chapter is that of fitting the solution to a differential equation to a set of experimental data and estimating a set of parameters associated with flie differential equation. The differential equation can be of either the initial value type or the boundary value type. A discussion of this general engineering problem has been postponed until both types of differential equations have been considered. One simple example of this comes from the field of Chemical Engineering where a model of the form ... 

The form of the Hamiltonian impedes efficient symplectic discretization. While symplectic discretization of the general constrained Hamiltonian system is possible using, e.g., the methods of Jay [19], these methods will require the solution of a nontrivial nonlinear system of equations at each step which can be quite costly. An alternative approach is described in [10] ( impetus-striction ) which essentially converts the Lagrange multiplier for the constraint to a differential equation before solving the entire system with implicit midpoint this method also appears to be quite costly on a per-step basis. 

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. 

Let x(t) and V(t) be the actual solutions to these differential equations. In general a given algorithm will replace these differential equations by a particular set of difference equations. These difference equations will then give approximate values of x(t) and V(t) at discrete, equally spaced points in time tu t2,. .., tn where tJ+x = tj + At. The differences between the solutions to the difference equations at tN and the solutions to the differential equations at t N depend critically on the time step At. If At is too large, the system of difference equations may be unstable or be in error due to truncation effects. On the other hand, if At is too small, the solutions to the difference equations may be in error due to the accumulation of machine rounding of intermediate results. 

Considerable progress has been made in recent years in obtaining solutions to the time-dependent Smith-Ewart differential difference equa> tions for various special types of reaction system in the nonsteady state. Although it has so far not proved possible to give an entirely general solution to these equations, it has proved possible to obtain a general solution to a modified set of equations which, under certain circumstances, approximate to the exact set of equations. 

The general solution of these differential equations can be derived for three special cases. To the best of our knowledge, this is the first time that a closed form solution given dependent Brownian motions within a USV model has been derived. 

Many of the fundamental equations in physics (and science in general) are formulated as differential equations. Typically, the desired mathematical function is known to obey some relationship in terms of its first and/or second derivatives. The task is to solve this differential equation to find the function itself. A complete treatment of the solution of differential equations is beyond the scope of this book, and only a simplified introduction is given here. Furthermore, we will only discuss solutions of differential equations with one variable. In most cases the physical problem gives rise to a differential equation involving many variables, but prior to solution these can often be (approximately) decoupled by separation of the variables, as discussed in Section 1.6. 

The integral of d/is / itself, while the integral of xdx is V2x/ except that any constant a can be added. This is completely general any first-order differential equation will give one additional integration constant that will have to be determined by some other means, for example from knowing the functional value at some point. That the found solution indeed is a solution to the differential equation can be verified by differentiation. 

The general solution of the differential equation (25) is a linear combination of the linearly independent solutions, where the constants of combination are determined by the initial conditions. In the special case considered below, from three to five terms of the asymptotic expansions in (26) and (27) are needed to compute fluxes to an accuracy of four decimal places. 

The questions so far (after necessary preliminaries included in (1), (2) and (3)) have dealt with direct problems. These problems have the general form given a differential equation what can we say about its solutions Another set of problems called inverse problems are even more important both in chemical kinetics and in general. It may even be said that solutions of direct problems only have practical importance when used to solve an inverse problem. 

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