Differential equation complex roots

In establishing the relationship between time-domain and Laplace-domain, we use only first and second order differential equations. That s because we are working strictly with linearized systems. As we have seen in partial fraction expansion, any function can be "broken up" into first order terms. Terms of complex roots can be combined together to form a second order term. 

Example 2.3 Complex Roots for an ODE Find the roots for the differential equation... 

Manometers and pressure springs may be described dynamically to a first approximation by second-order differential equations for which the roots of the characteristic equation are conjugate complex. As shown in Section III, 8, lc, since the roots are complex, these systems have an oscillatory mode, and the response of the system to step forcing, for example, is a damped sinusoid. 

Equation 3.19, Equation 3.29, and Equation 3.33 give the general solutions of the second order, constant coefficient, homogeneous, and linear differential equation for the respective cases of real unequal, repeated, and complex characteristic roots. However, the actual steps that are used in deriving a solution to the homogeneous problem are as follows ... 

We are now ready to discuss the treatment of differential equations with switching conditions. The main task is the localization of the switching points as roots of the switching functions. Note, the complexity of this task depends on the form of the switching function. Switching functions may in general have the same complexity as the right hand side function. For example, an impact requires the computation of relative distances for both functions. 

When the complexity of the mechanism is increased to a two step reaction, then the solutions of the Laplace forms of the equations involve the two roots of a quadratic as indicated above for second order differential equations. Recently Zhang, Strand White (1989) have suggested how a general matrix solution of rate equations in the Laplace form can be used to model kinetic mechanisms. Zhang et al. (1989) suggest this method as an alternative to numerical integration, but its use is, of course, restricted to linear equations like that of the more elegant matrix method described in section 4.2. 

In certain problems it may be necessary to locate all the roots of the equation, including the complex roots. This is the case in finding the zeros and poles of transfer functions in process control applications and in formulating the analytical solution of linear nth-order differential equations. On the other hand, different problems may require the location of only one of the roots. For example, in the solution of the equation of state, the positive real root is the one of interest. In any case, the physical constraints of the problem may dictate the feasible region of search where only a subset of the total number of roots may be indicated. In addition, the physical characteristics of ihe problem may provide an approximate value of the desired root. 

Depending on the sign and/or the magnitude of constants A and S, the roots to Equation (6-16) may be real, imaginary, or complex. Let us consider the solutions to the differential equation in each of these cases. 

The transient response for a differential equation has terms of the form exp(sjt) where is a root of the eharaeteristie equation. In another application, a linear system transfer funetion sueh as that ealeulated in the previous section and plotted in magnitude and angle in Figures 4.6 and 4.7 is known to be expressible as a ratio of numerator and denominator polynomials in the complex variable, s = jco. The roots of the denominator polynomial are again related to the transient response and to the stability of a system. These are but two examples of real world appheations for finding the roots of a polynomial of some order n. 

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