General solution of equation

It is essential that the roots are pure imaginary and this indicates that the solution is expressed in terms of sinusoidal functions. The general solution of Equation (3.85) is... 

Since the partial differential equation (2.6) is linear, any linear superposition of solutions is also a solution. Therefore, the most general solution of equation (2.6) for a time-independent potential energy V(x) is... 

Figure 8 reveals that the few data available for surfactant-laden bubbles do confirm the capillary-number dependence of the proposed theory in Equation 18. Careful examination of Figure 8, however, reveals that the regular perturbation analysis carried out to the linear dependence on the elasticity number is not adequate. More significant deviations are evident that cannot be predicted using only the linear term, especially for the SDBS surfactant. Clearly, more data are needed over wide ranges of capillary number and tube radius and for several more surfactant systems. Further, it will be necessary to obtain independent measurements of the surfactant properties that constitute the elasticity number before an adequate test of theory can be made. Finally, it is quite apparent that a more general solution of Equations 6 and 7 is needed, which is not restricted to small deviations of surfactant adsorption from equilibrium. 

In spite of the apparent simplicity of the method, its drawback comes from the fact that a two-spin system has been assumed. It provides merely global information spanning all protons prone two interact by dipolar coupling with the considered carbon. Selective information requires pulsed experiments stemming from the general solution of Equation (14) given below. 

In these circumstances the solutions y2(x) and y2 x) are linearly independent and the general solution of equation (3.1) is of the form... 

If v is an integer, say, then I n x) is a multiple of I (x) so that the solution (33.3) in effect contains only one arbitrary constant. By a process, similar lo that outlined in 30 we can show that in these circumstances the general solution of equation (31.2) is... 

The electronic equation is defined for a momentary position of the nuclei, and it has to be solved for various choices of Rf. As a consequence, the electronic energy depends in a parametric way on the nuclear positions and the energy curve (surface) El (R") is the general solution of Equation 5. 

The general solution of Equation 2-35, as well as a simple solution for the special case of f = 2 b, can be found in Box 2-1 (Zhang, 1994). For the special case of Ef = 2 b if the initial temperature is high so that the final speciation does not depend on the initial temperature, the final concentration of species A after complete cooling down of the system is... 

Now the behavior of the general solution of Equation 5-89b is examined. As t co, the volume-averaged concentration reaches a constant value (steady state) ... 

Since the constant A appears in both the numerator and denominator in the right side of equation (7.35), it can be cancelled to yield the general solution of equation (7.31) ... 

Bischoff, K. B. (1969) General solution of equations representing the effects of catalyst deactivation in fixed-bed reactors. Ind. Engng Chem. Fundam. 8, 665-668. 

The general solution of equation (33) in cylindrical coordinates can be written as a series of modes of the form [47]... 

The solution of Equation 14.16 with variable parameters can be obtained by application of the method used for biochemical reaction in the previous subsection. The general solution of Equation 14.16 is as follows, in the case of constant... 

This is a two-order linear ordinary differential equation. The general solution of Equation B. 13 can be written as... 

The general solution of equation (2-33) could be obtained using two classical relationships for partial derivatives [for simple forms of the distribution function i//(c)]... 

Under the assumption that the point pulse source is located at the origin of coordinates and generates a pulse at time t = 0, one can find a general solution of equation (13.64) in the form... 

The value of iT,a gives the normalizing factor for currents I = i/iT,The general solution of equations involving heterogeneous kinetics with respect to a tip above a conducting substrate can be obtained under reasonable boundary conditions in the form of two-dimensional integral equations (see Chapter 5). 

The general solution of Equation 9-1 is the sum of all the particular solutions with arbitrary coefficients. We consequently write as the general expression for the wave function for this system... 

Without loss of generality we set Ci = 1 because it merely serves as normalization. The second boundary condition fixes the still unknown eigenvalues A. They constitute an infinite countable set A due to the finiteness of d and b. Therefore, the general solution of equation (11.15) can be expressed... 

And the most general solution of Equation 2.8 is obtained by summing solutions of the type given by Equation 2.21 and a linear function ax + b ... 

The general solution of equation (4.71) for an isotropic solid shows the existence of waves that are purely longitudinal (compressive) and purely transverse (shear). In the more difficult case of an anisotropic solid, we consider the existence of a plane wave represented by... 

---