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There exists, consequently, a unique Green’s function. s/x 3 ; for s Ä x. 44) C1 D s 3 ; C2 D 6 2 2 6 is obtained immediately, because of the upper triangular form of the coefﬁcient matrix. 44). 40). 46) This example shows that even for higher-order equations, the procedure for the construction of Green’s functions utilized in this section is compact enough and requires only limited amount of work. If we examine the form of the Green’s functions we constructed so far in this section, we may notice a common property: they are symmetric in a certain sense.
In this case ﬁnding a fundamental set of solution is a routine procedure. It is worth noting that variable coefﬁcients do not decisively restrict the described algorithm, if a fundamental set of solutions is obtained in terms of either elementary or well-tabulated special functions. In other words, if the governing equation allows for an exact solution, one can readily construct a Green’s function by means of this algorithm. The following example illustrates this point. 6. 0; a/, it is important to impose certain limitations on the constant parameters m and b.
48) different from all the others considered earlier in this section? The reader will ﬁnd the reasoning behind this fact in the next section. 2 Symmetry of Green’s Functions In order to address the issue of symmetry for Green’s functions with respect to the observation and the source point, we have to carry out some preparatory work, which we will do in this section. x/; i D 0; n. p0 y/ 1 n 1d C . x/ D 0. x/ D 0 is called self-adjoint. x/ Á . a; b/: 2 dx dx We can make this limitation without loss of generality, but it markedly condenses it and makes it easier to comprehend.