Ordinary differential equations linear homogeneous

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

A non-linear mathematical model, which is a set of ordinary differential equations, for the process in the SPBER was developed.19 The model accounts for the heterogeneous electrochemical reaction and homogeneous reaction in the bulk solution. The lateral distributions of potential, current density and concentration in the reactor were studied. The potential distribution in the lateral dimension, x, of the packed bed was described by a one dimensional Poisson equation as ... 

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. 

Equation 6.1 through Equation 6.3 are partial differential equations (PDEs), as opposed to ordinary differential equations, but the deSnitimis of linearity and homogeneity remain the same as those given for secmid-order ordinary 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 ... 

Homogeneous liquid in a uniform linear core. The pressure partial differential equation governing transient, compressible, lineal, homogeneous, liquid flows having constant properties is 6 p(x,t)/9 = (( )pc/k) dpidi. Here p is pressure, while x and t represent space and time ( ), k, p, and e are rock porosity, rock permeability, fluid viscosity, and net fluid-rock compressibihty, respectively. If we assume a eonstant density, incompressible flow, and ignore the eompressibility of the fluid by setting c = 0, the right side of this equation identically vanishes. Then, the model reduces to the ordinary differential equation d p(x t)/d = 0 where t is a parameter as opposed to a variable. 

Here we show how the homogeneous system of linear ordinary differential equations, Eq. (VIII.6) can be solved by well-known methods of calculus (26,54). In the most general case Eq. (VIII.6) can be rewritten in matrix notation as... 

Rearrange this linear second-order homogeneous ordinary differential equation to obtain... 

A special class of homogeneous linear algebraic equations arises in the study of vibrating systems, structure analysis, and electric circuit system analysis, and in the solution and stability analysis of linear ordinary differential equations (Chap. 5). This system of equations has the form... 

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... 

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. 

Using assumption (4) and adopting the condition that the process is homogeneous in time (i.e transition probability function depends only on t - s and not specifically on / and s) the Kolmogorov equations (5.9) and (5.10) can be reduced to (linear) ordinary differential-difference equation ( differential in time, and difference in state and in accordance with (5.27) we get, for a deterministic initial condition... 

The most general constant coefficient, linear, second-order, ordinary, homogeneous differential equation is... 

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. 

A homogeneous ordinary linear differential equation with constant coefficients can be solved as follows ... 

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