Ordinary differential equations inhomogeneous equation

To solve a second-order inhomogeneous ordinary differential equation, either the Green s function method or the variation of parameters method can be used. Consider the self-adjoint equation... 

These ordinary differential equations may easily be solved by using the formulas for undamped inhomogeneous oscillations (Kamke, 1956) ... 

As we have already noted, the eigenvalues for the linearized deterministic system at the inhomogeneous fixed point, plotted as black dots in Figs. 5.25b,c are computed numerically for the system (5.30) in the deterministic case by using the spatially discretized set of ordinary differential equations. 

The discretization of the ordinary differential equation, Eq. (36), and of the two mentioned boundary conditions leads finally to a complete linear equation system whose inhomogeneity results from the discretized normalization condition, Eq. (35). An efficient resolution of this system becomes possible if those terms obtained by the parabolic interpolation are iteratively treated in the resolution procedure. 

This chapter focuses on the stability properties of networks or arrays of coupled monostable units or cells. We consider two types of coupling, namely diffusive coupling and photochemical coupling. The two main concerns are how the topology of the network connectivity and how spatial inhomogeneities in the array affect instabilities. Spatially discrete systems or networks of coupled cells are described by sets of ordinary differential equations. Methods to determine the stability of stationary states of ODEs are well developed. 

The model of isotope transfer in CSTR is a set of ordinary inhomogeneous differential equations with linear right side, which are solved by standard methods of expansion into eigenvectors. With the known eigenvalues and eigenvectors, it is easy to calculate first- and second-order derivatives, which makes the minimization procedure much faster. 

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