Homogeneous linear differential 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. 

This is a homogeneous linear differential equation of second order and its characteristic equation is... 

A homogeneous linear differential equation with constant coefficients can be solved by use of an exponential trial solution. 

The second assumption reduces Eqn. 25 to non-homogeneous linear differential equation with the following solution ... 

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. 

If the xk)t are known by solving (3.77), Eq. (3.80) is a system of non-homogeneous linear differential equations with given time dependent coefficients for the variance matrix Oik(t). 

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

Consider the system described by the linear, homogeneous ordinary differential equations... 

Equation (4.2.7) is a homogeneous system of linear differential equations. The search for a particular solution in the form... 

The Dimensionless Parameter is a mathematical method to solve linear differential equations. It has been used in Electrochemistry in the resolution of Fick s second law differential equation. This method is based on the use of functional series in dimensionless variables—which are related both to the form of the differential equation and to its boundary conditions—to transform a partial differential equation into a series of total differential equations in terms of only one independent dimensionless variable. This method was extensively used by Koutecky and later by other authors [1-9], and has proven to be the most powerful to obtain explicit analytical solutions. In this appendix, this method will be applied to the study of a charge transfer reaction at spherical electrodes when the diffusion coefficients of both species are not equal. In this situation, the use of this procedure will lead us to a series of homogeneous total differential equations depending on the variable, v given in Eq. (A.l). In other more complex cases, this method leads to nonhomogeneous total differential equations (for example, the case of a reversible process in Normal Pulse Polarography at the DME or the solutions of several electrochemical processes in double pulse techniques). In these last situations, explicit analytical solutions have also been obtained, although they will not be treated here for the sake of simplicity. 

SI or cgs units). The above matrix is equivalent to two linear homogeneous algebraic equations (the third and sixth equations) and four linear differential equations (the first, second, fourth, and fifth equations) the third equation is... 

A differential equation that cannot be put into this form is nonlinear. A linear differential equation in y is said to be homogeneous as well if R[x) 0, Otherwise, it is nonhomogeneous. That is, each term in a linear homogeneous equation contains the dependent variable or one of its derivatives after the equation is cleared of any common factors. The term i (jc) is called the nonhomogeneous term. 

This linear homogeneous partial differential equation represents the motion of stretched strings, the small oscillations of air in narrow (organ) pipes, and the motion of waves on the sea if the water is neither too deep nor too shallow. Let us now proceed to the integration of this equation. 

Equation 4.13 is a nonhomogeneous, linear differential equation. The solution can be written as the sum of what is called the particular solution and the solution to the homogeneous equation 12. One particular solution to the equation is the constant solution... 

To locate the marginal state, expressions such as Eq. (11) are substituted into Eqs. (8-10). The real part of q is then set equal to zero, and the resulting set of linear homogeneous ordinary differential equations is solved, subject to the appropriate boundary conditions which are also generally homogeneous. 

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

The general solution of linear differential equations is a linear combination of a homogeneous solution and a particular solution. For Eq. B.79, the particular solution is simply the steady-state solution that is... 

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... 

Standard techniques of vector analysis allow to equate the heat flow into the volume V to the heat flow across its surface. This operation leads to the linear and homogeneous Fourier differential equation of heat flow, given as Eq. (3). The letter k represents the thermal diffusivity in m s, which is equal to the thermal conductivity k divided by the density and specific heat capacity. The Laplacian operator is + d dy + d ld-z, where x, y, and z are the space coordinates. 

Here a denotes a state-velocity vector of the dependent variables of the system of n ordinary quasi-linear differential equations which constitute the model. For example, in the elementary (rather obsolete) homogeneous equilibrium model the components of vector a = h,P,w are conveniently chosen as enthalpy h, pressure P, and veloctiy w [4]. In two-fluid models the number of equations, n, may reach, or even exceed, six. 

With the dynamics of the initiator decomposition in the homogeneous phase taken into account, the light illumination gradient is described by a system of non-linear differential equations in partial derivatives ... 

Development-controlling prepattern mechanisms have been modelled in reaction-diffusion context. In the celebrated paper of Turing (1952) a model was presented in terms of reaction-diffusion equations to show how spatially inhomogeneous arrangements of material might be generated and maintained in a system in which the initial state is a homogeneous distribution. Two components were involved in the model, and the reactions were described by linear differential equations. The model in one spatial dimension s is ... 

Equation 3.19, Equation 3.29, and Equation 3.33 give the general solutions of the second order, constant coefficient, homogeneous, and linear differential equation for the respective cases of real unequal, repeated, and complex characteristic roots. However, the actual steps that are used in deriving a solution to the homogeneous problem are as follows ... 

One of the most common solution techniques applicable to linear homogeneous partial differential equation problems involves the use of Fourier series. A discussion of the methods of solution of linear partial differential equations will be the topic of the next chapter. In this chapter, a brief outline of Fourier series is given. The primary concerns in this chapter are to determine when a function has a Fourier series expansion and then, does the series converge to the function for which the expansion was assumed Also, the topic of Fourier transforms will be briefly introduced, as it can also provide an alternative approach to solve certain types of linear partial differential equations. 

Dividing the entire equation by AnDgAr and taking the limit as Ar approaches zero, a second-order linear homogeneous ordinary differential equation is obtained ... 

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