Higher-order differential equations

The higher-order differential equations, especially those of order 2, are of great importance because of physical situations describable by them. 

Equation (8-14) shows that starts from 0 and builds up exponentially to a final concentration of Kcj. Note that to get Eq. (8-14), it was only necessaiy to solve the algebraic Eq. (8-12) and then find the inverse of C (s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher order differential equations with Laplace transforms. 

Numerical solution of higher order differential equations is accomplished most conveniently by first converting them into an equivalent set of first order equations. Thus the second order equation... 

Modeling of some systems leads to higher order differential equations of the form... 

Any reaction scheme can be described by the set of first- or higher-order differential equations. These can be solved exactly for only relatively simple schemes. For more complicated sets, it is necessary to use either analog computers or digital numerical methods. 

The arbitrary sign in the last equations may be eliminated at the expense of ending up with a higher order differential equation [101]. Let us consider Eqn. (3.3.6) as an example, neglecting for a while the Brownian force X q,t) that will be reinstated later. If we time differentiate and multiply by Tq, then add the result to the starting equation to get rid of the memory integral, we may write ... 

The method described here is general and can be applied to higher-order differential equations. The method provides an attractive alternative to the use of particular solutions obtained using trial solutions based on the form of the function f x) and, in some cases, on the form of the homogeneous solution. ... 

Y. Bayazitoglu and J. Higenyi, Higher Order Differential Equations of Radiative Transfer Py Approximation, AIAA Journal, 14, p. 424,1979. 

In Section 9.4.1, selected numerical methods are examined for solving the initial value problems associated with first-order differential equations. Those methods are also applicable to higher-order differential equations following the reduction to a system of first-order equations. For example, the second-order differential equation... 

C.2 THE TRANSFORMATION OF HIGHER-ORDER DIFFERENTIAL EQUATIONS INTO A SET OF FIRST-ORDER DIFFERENTIAL EQUATIONS... 

With increasing velocity and energy the solitary waves become narrower and their width can be in the order of the lattice constant. In this case the QCA does not hold. And it would not help to take more terms of the Taylor expansion of ( ) in Eq. 16 into account. The resulting higher order differential equations cannot be integrated like Eq. 17. Moreover, even the infinite Taylor series cannot fully represent A q) because of the finite radius of convergence. 

Under the condition the Lagrangian (3. 474) be invariant for the transformation (3.468), i.e., equal with (3.473), the correction Lagrangian (3.475) should be canceled, that is the higher order differential equation is provided (Putz, 2008d)... 

Systems of coupled ODEs might arise from reformulating a higher-order differential equation to a system of first-order differential equations, or as a description of a system that consists of coupled variables. Systems of differential equations can be solved as an extention of the methodology for a single differential equation. The principle is shown in Example 6.4, which considers a stirred tank reactor. 

The methods described earlier in this chapter apply to higher-order differential equations,... 

Write the higher-order differential equation as a system of first-order differential... 

Next we apply the Laplace transform solution to a higher-order differential equation. 

In general, a transform expression may not exactly match any of the entries in Table 3.1. This problem always arises for higher-order differential equations, because the order of the denominator polynomial (characteristic polynomial) of the transform is equal to the order of the original differential equation, and no table entries are higher than third order in the denominator. It is simply not practical to expand the number of entries in the table ad infinitum. Instead, we use a procedure based on elementary transform building blocks. This procedure, called partial fraction expansion, is presented in the next section. 

In all the above examples, the systems were chosen so that the models resulted in sets of simultaneous first-order ordinary differential equations. These are the most commonly encountered types of problems in the analysis of multicomponent and/or multistage operations. Closed-form solutions for such sets of equations are not usually obtainable. However, numerical methods have been thoroughly developed for the solution of sets of simultaneous differential equations. In this chapter, we discuss the most useful techniques for the solution of such problems. We first show that higher-order differential equations can be reduced to first order by a series of substitutions. 

Many problems of interest involve systems of coupled differential equations. Thus before looking at coded solution techniques, approaches to handling systems of possibly coupled differential equations will be considered. As one example, it is possible to formulate a higher order differential equation in terms of a coupled system of first order differential equations. As an example consider a simple second-order differential equation of the form... 

---