General solution to a differential

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. 

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. 

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. 

This result is the recursion formula which allows the coefficient an+2 to he calculated from the coefficient a . Starting with either ao or a an infinite series can be constructed which is even or odd respectively. These two coefficients are of course the two arbitrary constants in the general solution of a second-order differential equation. If one of them, say ao is set equal to zero, the remaining series will contain the constant at and be composed only of odd powers of On the other hand if a 0, the even series will result. It can be shown, however that neither of these infinite series can be accepted as... 

If J(t) from Eq. 21-11 or 21-12 is inserted into Eq. 21-4, we get a linear differential equation with a time variable inhomogeneous term but constant rate k. The corresponding solution is given in Box 12.1, Eq. 8. Application of the general solution to the above case is described in Box 21.3. The reader who is not interested in the mathematics can skip the details but should take a moment to digest the message which summarizes our analytical exercise. 

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. 

Even not making any attempt to find a general solution to this rather complicated system, let us analyse the most important consequences resulting immediately from the differential equations and their solutions in a few simplest limiting cases. 

Let us, for a moment, consider a single particle in one dimension with a Hamiltonian of the type// = p2/2m + V(x). This is a second-order differential operator, and this means that the general solution to the inhomogeneous Eq. (3.51)—considered as a second-order differential equation—will consist of a linear superposition of two special solutions, where the coefficients will depend on the boundary conditions introduced. As a specific example, one could think of the two solutions to the JWKB problem, their connection formulas, and the Stoke s phenomenon for the coefficients. 

Briefly, the aim of Lie transformations in Hamiltonian theory is to generate a symplectic (that is, canonical) change of variables depending on a small parameter as the general solution of a Hamiltonian system of differential equations. The method was first proposed by Deprit [75] (we follow the presentation in Ref. 76) and can be stated as follows. 

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. 

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