Linear First-Order Differential Equation

Linear Equations A differential equation is said to be linear when it is of first degree in the dependent variable and its derivatives. The general linear first-order differential equation has the form dy/dx + P x)y = Q x). Its general solution is... 

Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

Equation (8.4.3) is a linear first-order differential equation of concentration and reactor length. Using the separation of variables technique to integrate (8.4.3) yields... 

This section is a review of the properties of a first order differential equation model. Our Chapter 2 examples of mixed vessels, stined-tank heater, and homework problems of isothermal stirred-tank chemical reactors all fall into this category. Furthermore, the differential equation may represent either a process or a control system. What we cover here applies to any problem or situation as long as it can be described by a linear first order differential equation. 

This is a linear, first-order differential equation, the solution of which is... 

For each occupied orbital y = 1,..., n, we have to solve the set of N -h 1 linear first-order differential equations (21). Unfortunately, it does not seem possible to obtain analytical solutions for any realistic choice of F(t) and, therefore, it is necessary to resort to finding approximate or numerical... 

The mathematical problem posed is the solution of the simultaneous differential equations which arise from the mass-action treatment of the chemistry. For the homogeneous, well-mixed reactor, this becomes a set of ordinary, non-linear, first-order differential equations. For systems that are not... 

This is an introductory book. The pace is leisurely, and where needed, time is taken to consider why certain assumptions are made, to discuss why an alternative approach is not used, and to indicate the limitations of the treatment when applied to real situations. Although the mathematical level is not particularly difficult (elementary calculus and the linear first-order differential equation is all that is needed), this does not mean that the ideas and concepts being taught are particularly simple. To develop new ways of thinking and new intuitions is not easy. 

Thus we have two simultaneous linear first-order differential equations in x and y,... 

This is a linear first-order differential equation that can be solved for [AM](t), and we can proceed to solve each of these successive Hnear differential equations for [AMj]. Without doing this, we can find the number average polymer length ii quite simply by noting the definition... 

What does time to steady-state mean Why is there no unique definition of time to steady-state for a linear first-order differential equation ... 

After combining eqs. (3.4) and (3.5) a set of simultaneous linear first order differential equations is obtained for the variables y. It reads ... 

Unfortunately, unlike the general linear first-order differential equation (7.31), there is no simple template which provides the solution, and we need therefore to apply different methods to suit the equation we meet in the chemical context. Equations of the general form given in equation (7.45) crop up in all branches of the physical sciences where a system is under the influence of an oscillatory or periodic change. In chemistry, some of the most important examples can be found in modelling ... 

Finding general solutions to linear first order differential equations using the integrating factor method. 

The controller scheme developed in the following is based on the well-known GMC paradigm [22, 27] reviewed in Sect. 5.4.2. The key idea of this technique is that of globally linearizing the reactor dynamics by acting on the jacket temperature 7], which is, in turn, controlled by a standard linear (e.g., PID) controller. Since 7] does not play the role of the input manipulated variable, the only way to impose an assigned behavior to the jacket temperature is that of computing a suitable setpoint 7j,des, to be passed by a control loop closed around 7], Both in [22] and [27], the mathematical relationship between the jacket temperature and the setpoint is assumed to be a known linear first-order differential equation, from which Tj es is... 

The constitutive relation is expressed as a linear first-order differential equation ... 

The denominator of this linear first-order differential equation gives the process system time constant of 20 min in the expression 1 + 20q. Likewise, the numerator gives the zero-frequency process gain of 15°F/(lb)(in2). 

The radiation balance of a layer with the thickness d having an infinitely large surface, irradiated homogeneously from one side with exciting radiation, is given by the solution of four coupled linear first-order differential equations (Eqs. 3.5-1...4). This is a boundary value problem, with the definitions given in Fig. 3.5-2. We are discussing... 

Solution of Eqns. (37-40) can be obtained by simultaneous integration of the resulting 2(Nq+N [) coupled linear first-order differential equations by difference techniques as presented in Section V. For time-independent Hamiltonians H the total energy... 

Example 2.1 Linear First-Order Differential Equation In the development of boundary-layer mass transfer to a planar electrode, a similarity transformation variable (see Section 2.4) can be identified through solution of... 

Remember 2.1 The general solution to nonhomogeneous linear first-order differential equations can be obtained as the product of function to be determined and the solution to the homogeneous equation. 

Equation (2.55) is a linear first-order differential equation. It can be solved by use of a variable transformation h = such that... 

Example 2.1 Linear First-Order Differential Equation 24... 

The general solution (3-6) to a set of linear first order differential equations such as Eqs. (5) is well known it is... 

Equation (6.12) is a system of 2N linear first order differential equations and can be written in matrix form as... 

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