Nontrivial solution

This results iu four equations and four unknowns. Siuee the equations are homogeneous, a nontrivial solution exists only if die detenuiuaut fonued by the eoeflfieieuts of A, B, C and D vanishes. The solution to this equation is... 

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

This solution for (X) in which all the unknowns are zero is called the trivial solution. A nontrivial solution to Equation (A.27) exists, therefore, only when matrix [A] is singular, that is, when A = 0. 

These equations are satisfied if 7a = 0, 7z = 0, but this trivial solution is of no interest. To ensure a nontrivial solution, it is sufficient to require that the determinant of the coefficients of 7a and 7z be equal to zero, namely. 

The coefficient determinant must vanish if this system is to have a nontrivial solution. This condition gives... 

Nontrivial solutions will exist only if the determinant D... 

This equation can always be satisfied with Xout = 0 so that the washout condition is always possible as a steady state. This steady state is achieved when there is no inoculum or when the flow rate is too high. A nontrivial solution with X ut > 0 requires that Fq, = 1 or... 

The nontrivial solutions of this problem, that is, the eigenfunctions Uj. and the appropriate eigenvalues Xj. are expressed by 1. 

Since only nontrivial solutions are those to be found, that is, sin ax 0, it follows from the foregoing that... 

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... 

This set of equations for the elements of L can be resolved by application of Cramer s rale. Then, a nontrivial solution exists only if the determinant of die coefficients vanishes, or... 

Equation (125) applies for all values of the index k — 1,2,..., m. It is a set of m simultaneous, homogeneous, linear equations for the unknown values of the coefficients c . Following Cramer s rule (Section 7.8), a nontrivial solution exists only if the determinant of the coefficients vanishes. Thus, the secular determinant takes the form... 

If the unit matrix E is of order n, Eq. (67) represents a system of n homogeneous, linear equations in n unknowns. They are usually referred to as the secular equations. According to Cramer s rule [see (iii) of Section 7.8], nontrivial solutions exist only if the determinant of the coefficients vanishes. Thus, for the solutions of physical interest,... 

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... 

Nontrivial solutions are obtained only if the secular equation... 

It can be shown that a nontrivial solution for a pair of linear equations requires that the determinant of the coefficients must be equal to 0. This means that... 

The determinant of (A - el) must be zero for a nontrivial solution (v 0) to exist. Let us illustrate this idea with a (2 X 2) matrix ... 

Equation (A.30) determines values of e which yield a nontrivial solution. Factoring (A.30)... 

Zeros of its determinant determine the effective refractive indexes of the eigenmodes, and from the corresponding nontrivial solutions p (77 ) we can calculate the eomplete vectorial field distribution everywhere in the cross-section using Eqs. (36) - (40) or their more stable equivalents. 

Similarly as before, nontrivial solutions to this equation correspond to eigenmodes of the bent waveguide. 

Figure 3. Nontrivial solution of a degenerate Riemann problem with initial data in the metastable area.

This problem can be solved by the method of separation of variables. The eigenvalue problem for the difference Laplace operator Ay = ySlXl + Vx2x2 supplied by the first kind boundary conditions may be set up in a quite similar manner as follows it is required to find the values of the parameter A (eigenvalues) associated with nontrivial solutions of the homogeneous equation subject to the homogeneous boundary conditions... 

So, problem (16a) has nontrivial solutions y = Tk 0, where Tk can be recovered from the equation... 

A set of n linear homogeneous equations in n unknowns always has the solution x, — x2 = xn = 0, which is the trivial solution. Suppose the coefficient determinant det(aiy) is not equal to zero for a set of linear homogeneous equations we can then use Cramer s rule (1.82). Since the equations are homogeneous, the determinant Rk will have a column of zeros, and will equal zero hence xt = x2= =xn = 0, and we have only the trivial solution. Thus for a nontrivial solution of the homogeneous equations to exist, we must have det(o,y) = 0. This condition can also be shown to be sufficient to insure the existence of a nontrivial solution. A system of n simultaneous, linear, homogeneous equations in n unknowns has a nontrivial solution if and only if the determinant of the coefficients equals zero. 

For a nontrivial solution of the equations (1.187), the coefficient determinant must vanish (Section 1.2). Hence... 

---