Exercise 10.4 Show that (10.32) follows from (10.31), when we take N D 2.

If we write the PDE (10.31) in the form

then a20 D 1; a11 D .v1 Cv2/; a02 D v1v2; a10 D q1 Cq2; a01 D v1q2 Cv2q1; a0 D 0.
Then, a short calculation shows that (i) The polynomial 2 Ca11Ca02 D 0 has two distinct real roots v1 (ii) a10 > 0.
(iii) v1 Conversely, one may characterize the coefficients as follows: suppose that we are given a second-order constant coefficient linear PDE written in the form (10.33) and satisfying the conditions (i)-(ii)-(iii). Then, there exist constants v1 0;q2 >
0 such that (10.32) holds.

Transcribed Image Text:

HuaDDu= 0, k+12