First-order linear

Variations of a continuous function over this element can be represented by a complete first-order (linear) polynomial as... 

It has been postulated that jet breakup is the result of aerodynamic interaction between the Hquid and the ambient gas. Such theory considers a column of Hquid emerging from a circular orifice into a surrounding gas. The instabiHty on the Hquid surface is examined by using first-order linear theory. A small perturbation is imposed on the initially steady Hquid motion to simulate the growth of waves. The displacement of the surface waves can be obtained by the real component of a Fourier expression ... 

Although the differential equation is first-order linear, its integration requires evaluation of an infinite series of integrals of increasing difficulty. 

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. 

Equation 3-133 is a first order linear differential equation of the form dy/dx -i- Py = Q. The integrating factor is IF = and... 

Equation (A4) is a first order, linear, ordinary differential equation which can be solved analytically for [PJ assuming X, and X, are constant over a small increment in time. Solving for [PJ from some time ti to tj gives Equation (1) (1). When X, is considered a function of time (i.e., initiator concentration is allowed to vary through the small time increment) while maintaining X, constant over the increment. Equation (A4) can again be solved analytically to give Equation (3). 

These equations form a set of first order linear differential equations with constant coefficients and with initial conditions ... 

The linearisation of the non-linear component and energy balance equations, based on the use of Taylor s expansion theorem, leads to two, simultaneous, first-order, linear differential equations with constant coefficients of the form... 

Quoting only first order (linear) age errors when the age plus age-error approaches the secular-equilibrium limit ... 

This first-order linear differential equation may be solved using an integrating factor approach to give... 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. 

This approach assumes that fe is known, the change in CL and k are proportional to CLcr, renal disease does not alter drug metabolism, any metabolites are inactive and nontoxic, the drug obeys first-order (linear) kinetic principles, and the drug is adequately described by a one-compartment model. The kinetic parameter/dosage adjustment factor (Q) can be calculated as ... 

Variables separable and first order linear are most often encountered. Exercises dealing with first order equations are in problem Pi.05.05. 

The integrating factor of this first order linear equation is exp(k2t) so the solution is,... 

Equation (2) is not needed for evaluation of the specific rates. It is a first order linear equation that could be integrated with the found specific rates and the resulting (B, t) could be compared with the given data for consistency. 

This is a first order linear equation with integrating factor, exp(k2t) and solution... 

Rearrange into a standard form of first order linear equation,... 

Conditions are C2 = C20 when t - 0. Substitute for Ca from Eq (1) and solve as a first order linear equation, or solve both equations numerically. 

The solution of this first order linear differential equation with B = 0 fc = 0 is... 

The above equation is a first-order linear differential equation which on solution gives... 

J.1 First-Order Linear Ordinary Differential Equation... 

Example 6A An isothennal, constant-holdup, constant-throughput CSTR with a first-order irreveraible reaction is described by a component continuity equation that is a first-order linear ODE ... 

The first-order linear equation [Eq. (6.44)] could have a time-variable coefficient that is, 0) could be a function of time. We will consider only linear second-order ODEs that have constant coefficients (tj, and ( are constants). 

VHien this method is used, Table II shows the results when the regression model is the normal first order linear model. Since the maximum absolute studentized residual (Max ASR) found, 2.29, was less than the critical value relative to this model, 2.78, the conclusion is that there are no inconsistent values. 

In general, taking the ratio of two rate equations eliminates the time variable and gives information on the product distribution. So dividing Eq. 34 by Eq. 32 we obtain the first-order linear differential equation... 

The coefficients of equation (5) were determined by stepwise multiple regression in which the tracer element accounting for the greatest proportion of the variation of [POM] is used to find a first order, linear regression equation of the form [POM]... 

The First-Order Linear Inhomogeneous Differential Equation (FOLIDE) First-Order Reaction Including Back Reaction Reaction of Higher Order Catalyzed Reactions... 

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