Homogeneous equation

Homogeneous equations comprise a third group of nonlinear-type problems that usually do not yield to either the variable-separable or exact solution techniques. An equation of this type, however, may yield a solution if a new variable can be introduced. 

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Any linearly independent set of simultaneous homogeneous equations we can construct has only the zero vector as its solution set. This is not acceptable, for it means that the wave function vanishes, which is contrai y to hypothesis (the electron has to be somewhere). We are driven to the conclusion that the normal equations (6-38) must be linearly dependent. 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

An eigenvalue problem is a homogeneous equation of the second land, and solutions exist only for certain A. 

Derive Equation (5.81). Hint for a set of homogeneous equations to have a nontrivial solution, the determinant of the coefficients must be zero. 

If and are linearly independent solutions of the homogeneous equation (44), then its general solution can be designed as a linear combination of and 2/p with arbitrary constants Cj and Cji... 

In such a setting it is required to find the values of the parameter A such that these homogeneous equations have nontrivial solutions y(x) 0. In contrast to the first boundary-value problem, here the parameter A enters not only the governing eqnation, but also the boundary conditions. The introduction of new sensible notations... 

That is to say, the meaning of stability of scheme (21) is that a solution (21) depends continuously on the right-hand side and this dependence is uniform in the parameter h. This implies that a small change of the right-hand side results in a small change of the solution. If the scheme is solvable and stable, it is correct. Note that the uniqueness of the scheme (21) solution is a consequence of its solvability and stability and, hence, we might get rid of the uniqueness requirement in condition (1). Indeed, assume to the contrary that there were two solutions to equation (21), say and By the linearity property of the operator A, their difference = y — yf should satisfy the homogeneous equation... 

Corollary 2 The homogeneous equation (1) subject to the boundary condition... 

Before giving further motivations, let us represent the solution of problem (16) as a sum y = y + y, where y is a solution of the homogeneous equation... 

Inequality (12) expresses the property of continuous dependence which is uniform in h and t of the Cauchy problem (4) upon the input data. Here and below the meaning of this property is stability. A difference scheme is said to be absolutely stable if it is stable for any r and h (not only for all sufficiently small ones). It is fairly common to distinguish the notion of stability with respect to the initial data and that with respect to the right-hand side. Scheme (4) is said to be stable with respect to the initial data if a solution to the homogeneous equation... 

The main goal of our studies is to find out sufficient conditions for the stability of scheme (1) and obtain a priori estimates for a solution of problem (1) expressing the stability of this scheme with respect to the right-hand side and the initial data. In preparation for this, a solution of problem (1) can be written as a sum y = y + y, where y is a solution to the homogeneous equation with the initial condition y(0) = j/(0) = y ... 

In Section 1 of the present chapter we have established that the residual = Ayj. — f satisfies the homogeneous equation... 

The same applies to the other eigenvectors U2 and Vj, etc., with additional constraints of orthonormality of u, U2, etc. and of Vj, Vj, etc. By analogy with eq. (31.5b) it follows that the r eigenvalues in A must satisfy the system of linear homogeneous equations ... 

These simultaneous linear homogeneous equations determine c and C2 and have a non-trivial solution if the determinant of the coefficients of c, C2 vanishes... 

Hence, as is often stated, the determination of the normal coordinates is equivalent to the successful search for a matrix L that diagonalizes the product GF via a similarity transformation. This system of linear, simultaneous homogeneous equations can be written in the form... 

Equations (16) and (17) form a pair of simultaneous, homogeneous equations. They cannot be solved uniquely for the components of C. However, their solution can be found in terms of a parameter a. The result, which can be easily verified (problem 1), is... 

A second question arises for those who understand the importance of dimensional analysis, a subject that is treated briefly in Appendix II. If A and B are both vector quantities with, say, dimensions of length, how can their cross product result in a vector C, presumably with dimensions of length The answer is hidden in the homogeneous equations developed above [Eqs. (IS) to (20)]. The constant a was set equal to unity. However, in this case it has the dimension of reciprocal length. In other words, C = aABsirtd is the length of the vector C. In general, a vector such as C which represents the cross product of two ordinary vectors is an areal vector with different symmetry properties from those of A and B. 

This result is a system of simultaneous linear, homogeneous equations for the coefficients, cu. Cramer s rule states that a nontrivial solution exists only if... 

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