Linear first-order equations

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... 

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

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. 

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). 

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... 

In Eqs. 12-47/48 we recognize the first-order linear inhomogeneous differential equation (FOLIDE, Box 12.1). Depending on whether the input I and the different rate constants (kw, kj) are constant with time, their solutions are given in Eqs. 6,8, or 9 of Box 12.1. 

Box21.6 Solution of Two Coupled First-Order Linear Inhomogeneous Differential Equations (Coupled FOLIDEs)... 

As a last step, a first-order (linear) reaction is added to the advective-difiusive equation of a sorbing substance, Eq. 25-39 ... 

Substitution of Equation 2 into Equation 1 results in the following first-order linear differential equation for w in terms of the relative axial distance z = x/L, where L is the... 

A first-order linear differential equation has the general form ... 

This is a first-order linear differential equation and can be solved using the method of integrating factors that we show below. Multiplying both sides by e1 2, we have... 

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