Linear first-order differential

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

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. 

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... 

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... 

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... 

With all of the above assumptions the Maxwell-Stefan relations (Eq. 2.1.16) reduce to a system of first-order linear differential equations... 

For a closed, isolated system H is time independent time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem. If (to) is known, a formal solution to Eq. (1) is given by... 

Fortunately, process control problems are most usually concerned with maintaining operating variables constant at particular values. Most disturbances to the process involve only small excursions of the process variables about their normal operating points with the result that the system behaves linearly regardless of how nonlinear the descriptive equations may be. Thus Eq. (1) is a nonlinear differential equation since both Cp and U are functions of 80 but for small changes in 8 average values of CP and U may be regarded as constants, and the equation becomes the simplest kind of first order linear differential equation. 

In this subsection, some commonly used numerical schemes that involve difference equations to solve ordinary differential equations are presented along with their stability characteristics. Simple examples to illustrate the effects of step size on the convergence of numerical methods are shown. A simple discretization of the first-order linear differential equation... 

These simple properties of the Laplace transform make it a very convenient tool for solving systems of first-order linear differential equations, such as the equations for growth and decay of nuclides in radioactive disintegrations and neutron inadiation. They permit these differential equations to be treated as if they were systems of simple transformed linear equations without derivatives. 

The response of a nonlinear system at steady state (see above) to a small perturbation is linear, i.e., describable as a set of first order linear differential equations (of the form Eq. (2)). Furthermore, kinetic models represented by such a set of linear differential equations contain many helpful pieces of information concerning the host nonlinear system, viz., the number of exchanging (metabolic) pools, the rate of exchange among the (metabolic) pools, and the size of the (metabolic) pools. 

Equation (4.11) is an inhomogeneous first-order linear differential equation of a t). By multiplying both sides of Eq. (4.11) by an integration factor exp(t/r), the Maxwell equation can be easily transformed into the integration form... 

Equation (6.57) is an inhomogeneous first-order linear differential equation of the tensor X[oj in the range —oo < t < t. With the condition that X[o] is finite at t = —oo, we may obtain the solution for X[oj. X[oj(t,t) = x(i), as the convected coordinates coincide with the fixed coordinates when... 

The integration result of this first-order linear differential equation is well known and is represented in general form without specification of n(t) [Eq. (18) in Ref 39]. An interesting peculiarity of this important derivation is the disappearance of terms, related to coagulation at the transition, from the equation set [Eq. (37)] to the main Eq. (38). This corresponds to the fact that the total quantity of droplets does not change due to coagulation it decreases due to coalescence only. 

This compact notation reminds us of some of the notions about first-order linear differential equations that we have used. For example, the notion of existence and uniqueness still applies. 

R, two linear algebraic equations in terms of the six components of T are found. Two components may thus be expressed in terms of the other four, resulting in four first-order linear differential equations for the other four variables, and two linear algebraic equations. These differential equations may be written in 4x4 matrix form as... 

This is a first-order linear differential equation with the initial condition at t = 0 [RR] = 0, giving [/ ] as... 

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