Differential equations solution with constant coefficients

The procedure we followed in the previous section was to take a pair of coupled equations, Eqs. (5-6) or (5-17) and express their solutions as a sum and difference, that is, as linear combinations. (Don t forget that the sum or difference of solutions of a linear homogeneous differential equation with constant coefficients is also a solution of the equation.) This recasts the original equations in the foiin of uncoupled equations. To show this, take the sum and difference of Eqs. (5-21),... 

Neither of these equations tells us which spin is on which electron. They merely say that there are two spins and the probability that the 1, 2 spin combination is ot, p is equal to the probability that the 2, 1 spin combination is ot, p. The two linear combinations i i(l,2) v /(2,1) are perfectly legitimate wave functions (sums and differences of solutions of linear differential equations with constant coefficients are also solutions), but neither implies that we know which electron has the label ot or p. 

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

Systems of reversible first-order reactions lead to sets of simultaneous linear differential equations with constant coefficients. A solution may be obtained by means of a matrix formulation that is widely used in quantum mechanics and vibrational... 

The preceding two equations are examples of linear differential equations with constant coefficients and their solutions are often found most simply by the use of Laplace transforms [1]. 

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

We have obtained an inhomogeneous differential equation with constant coefficients. As follows from the theory of linear equations, its solution is a sum... 

In this section, we will outline only those properties of the Laplace transform that are directly relevant to the solution of systems of linear differential equations with constant coefficients. A more extensive coverage can be found, for example, in the text book by Franklin [6]. 

Linear Differential Equations with Constant Coefficients and Right-Hand Member Zero (Homogeneous) The solution of y" + ay + by = 0 depends upon the nature of the roots of the characteristic equation mr + am + b = 0 obtained by substituting the trial solution y = emx in the equation. 

This is a second-order differential equation with constant coefficients. A solution of the form... 

Exercise. Show that dtP x, t) = a does not tend to a stationary solution when a > 0 and does not conserve positivity when a < 0. Extend this conclusion to general differential equations with constant coefficients. 

Find the general solutions to linear second-order differential equations with constant coefficients by substitution of trial functions... 

The equilibrium equations of a hyperboloid of revolution used for cooling towers derived by using membrane theory under an arbitrary static normal load are reduced to a single partial differential equation with constant coefficients. The problem of finding displacements is reduced to a similar type of equation so that the solution for this problem becomes straightforward. 11 refs, cited. 

Equation 1.12 is a system of linear differential equations with constant coefficients. Then, following the rules for solving this type of an equation, its solution can be written in the following form [7] ... 

Equation 3-56 is a linear, homogeneous, second-order differential equation with constant coefficients. A fundamental theory of differential equations states (hat such an equation has two linearly independent solution functions, and its general solution is the linear combination of those two solution functions. A careful examination of the differential equation reveals that subtracting a constant multiple of the solution function 0 from its second derivative yields zero. Thus we conclude that the function 0 and its second derivative must be constant multiples of each other. The only functions whose derivatives are constant multiples of the functions themselves are the exponential functions (or a linear combination of exponential functions such as sine and cosine hyperbolic functions). Therefore, the solution functions of the differential equation above are the exponential functions e or or constant multiples of them. This can be verified by direct substitution. For example, the second derivative of e is and its substitution into Eq. 3-56... 

Remember 2.2 The general solution to nonhomogeneous linear second-order differential equations with constant coefficients can be obtained as the product of function to be determined and the solution to the homogeneous equation (see equation (2.41)). 

In the course of developing models for the impedance response of physical systems, differential equations are commonly encountered that have complex variables. For equations with constant coefficients, solutions may be obtained using the methods described in the previous sections. For equations with variable coefficients, a numerical solution may be required. The method for numerical solution is to separate the equations into real and imaginary parts and to solve them simultaneously. 

Solution Equation (2.70) is a linear second-order homogeneous differential equation with constant coefficients. It can be solved using the characteristic equation... 

Since c, the Nj and the D,y are constant, Eqs. 8.3.7 represents a set of linear differential equations with constant coefficients. The solution, in — 1 dimensional matrix form, is (see Appendix B.2 for derivation)... 

Equation 13.3.14 is a first-order matrix differential equation with constant coefficients (we have already assumed that [E] and [A/] are constant matrices). The solution is... 

Recall that linear differential equations with constant coefficients accept exponential solutions. 

A homogeneous linear differential equation with constant coefficients can be solved by use of an exponential trial solution. 

As mentioned earlier, the primary use for the Laplace transforms is to solve linear differential equations or systems of linear (or linearized nonlinear) differential equations with constant coefficients. The procedure was developed by the English engineer Oliver Heaviside and it enables us to solve many problems without going through the troubteof>tr finding the complementary and the particular solutions for linear differential equations. The same procedure can be extended to simple or systems of partial differential equations and to integral equations. 

Certain types of differential equations occur sufficiently often to justify the use of formulas for the corresponding particular solutions. The following set of Tables I to XIV covers all first, second, and th order ordinary linear differential equations with constant coefficients for which the right members are of the form F[x)e sinsx or P x)e cossx, where r and s are constants and P(x) is a polynomial of degree n. 

When the right member of a reducible linear partial differential equation with constant coefficients is not zero, particular solutions for certain types of right members are contained in Tables XV to XXL In these tables both F and P are used to denote polynomials, and it is assumed that no denominator is zero. In any formula the roles of x and y may be reversed throughout, changing a formula in which x dominates to one in which y dominates. Tables XIX, XX, XXI are applicable whether the equations are reducible or... 

In most applications of the theory to date, the solution of the Redfield equation has required first the explicit calculation of the Redfield tensor elements [Eq. (11)] given these, Eq. (10) could be solved as an ordinary set of linear differential equations with constant coefficients, either by explicit time stepping [41, 42] or by diagonalization of the Redfield tensor [37,38]. Since there are such tensor elements for an A -state subsystem, the number of these quantities can become quite large. Because of this, until recently most applications of Redfield theory have been limited to small systems of two to four states, or else assumptions, such as the secular approximation, have been used to neglect large classes of tensor elements. 

---