Homogeneous Linear Equations with Constant Coefficients

1 Homogeneous Linear Equations with Constant Coefficients 

The mass balance for a first-order reaction in a tubular reactor with a flow velocity of v and the concentration of reactant in feed, Cp undergoing a reaction with the rate constant of k is governed by the following second-order linear equation with constant coefficients  

Consider an inhomogeneous linear differential equation with constant coefficients given by 

For C(x) to assume the same functional form as if a and P were constants, it is required that 

---

The procedure we followed in the previous section was to take a pair of coupled equations, Eqs. (5-6) or (5-17) and express their solutions as a sum and difference, that is, as linear combinations. (Don t forget that the sum or difference of solutions of a linear homogeneous differential equation with constant coefficients is also a solution of the equation.) This recasts the original equations in the foiin of uncoupled equations. To show this, take the sum and difference of Eqs. (5-21),... 

Solution Equation (2.70) is a linear second-order homogeneous differential equation with constant coefficients. It can be solved using the characteristic equation... 

A linear homogeneous differential equation with constant coefficients can be solved by the following routine method ... 

Eqs. (2.40) and (2,41) are systems of s linear homogeneous differential equations with constant coefficients for all the s unknown and Aa,- respectively. These systems of equations are equivalent to a single differential equation of jth order for a single unknown [17]. Taking eq. (2.41), this differential equation for the tth concentration is given by... 

So far we have considered only cases where the potential energy V(ac) is a constant. Hiis makes the SchrSdinger equation a second-order linear homogeneous differential equation with constant coefficients, which we know how to solve. However, we want to deal with cases in which V varies with x. A useful approach here is to try a power-series solution of the SchrSdinger equation. 

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

Linear Differential Equations with Constant Coefficients and Right-Hand Member Zero (Homogeneous) The solution of y" + ay + by = 0 depends upon the nature of the roots of the characteristic equation mr + am + b = 0 obtained by substituting the trial solution y = emx in the equation. 

Equation 3-56 is a linear, homogeneous, second-order differential equation with constant coefficients. A fundamental theory of differential equations states (hat such an equation has two linearly independent solution functions, and its general solution is the linear combination of those two solution functions. A careful examination of the differential equation reveals that subtracting a constant multiple of the solution function 0 from its second derivative yields zero. Thus we conclude that the function 0 and its second derivative must be constant multiples of each other. The only functions whose derivatives are constant multiples of the functions themselves are the exponential functions (or a linear combination of exponential functions such as sine and cosine hyperbolic functions). Therefore, the solution functions of the differential equation above are the exponential functions e or or constant multiples of them. This can be verified by direct substitution. For example, the second derivative of e is and its substitution into Eq. 3-56... 

The general homogeneous linear second-order di fferential equation with constant coefficients can be written as... 

Remember 2.2 The general solution to nonhomogeneous linear second-order differential equations with constant coefficients can be obtained as the product of function to be determined and the solution to the homogeneous equation (see equation (2.41)). 

This equation is a linear homogeneous equation with constant coefficients, so a trial solution of the form of Eq. (8.14) will work. The characteristic equation is... 

Linear Differential Equations with Constant Coefficients and Right-Hand Member Zero (Homogeneous) The solution of... 

Equation (2.3) describes a set of linear homogeneous simultaneous differential equations with constant coefficients, which is readily solved once the eigenvalues and eigenvectors of Q are known. Let the matrix Q be diagonalised by the transformation... 

The problem is separable for a bare homogeneous reactor. However, only the case of a step input of reactivity, i.e., the case of a constant value of p, is easily solved. In this case, the kinetic equations are readily reduced to a second order (for the case of one delayed neutron group) homogeneous linear differential equation with constant coefficients. For an input of positive reactivity two solutions arise, of the form and where o>i > 0 and 0)2 < 0. The first solution controls the persisting exponential rise of the flux, where it is recalled that T = l/o>i is the reactor period, and the second solution which rapidly becomes small is called the transient solution. 

In the limit as Ax approaches zero, a second-order linear homogeneous equation with constant coefficients is obtained ... 

An important special case is a linear homogeneous second-order differential equation with constant coefficients ... 

For the special case of a linear differential equation with constant coefficients (i.e., the, ) in Equation (21) are constants), the procedure for finding the homogeneous solution is as follows ... 

This is a homogeneous ordinary linear differential equation with constant coefficients. We say that it is second order, which means that the highest order derivative in the equation is a second derivative. 

Depending on their origin, difference equations may be linear or nonlinear, homogeneous or nonhomogeneous, with constant or variable coefficients. For the purposes of this book, it will be necessary to discuss only the methods of solution of homogeneous linear difference equations with constant coefficients. 

It can be concluded from the above discussion that the solution of homogeneous linear difference equations with constant coefficients is of the form... 

---