By H. F. Weinberger

Textual content offers the final houses of partial differential equations reminiscent of features, domain names of independence, and greatest rules. ideas.

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Extra resources for A First Course in Partial Differential Equations: with Complex Variables and Transform Methods (Dover Books on Mathematics)

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4. Given 1 0 A= 1 2 1 0 0 0 -1 . Compute the 3 x 3 matrix eAt and solve x = Ax. Cf. Problem 2 in Problem Set 2. 5. Find the solution of the linear system is = Ax where (a) A = 120 -121 2 (b) A = [1 2] 10 0, (c) A = -2 (d) A = 0 1 -2 L0 1 0 0 -2 . 1. Linear Systems 20 6. , for all x E E, T(x) E E) and let T(x) = Ax with respect to the standard basis for R". Show that if x(t) is the solution of the initial value problem is = Ax x(0) = xo with xO E E, then x(t) E E for all t E R. 7. Suppose that the square matrix A has a negative eigenvalue.

Al = A2 = 2. Thus, 10 2 S- 0 2] and N=A-S= 1 ' 1 It is easy to compute N2 = 0 and the solution of the initial value problem for (1) is therefore given by x(t) = eAtxo = e2t[I + Nt]xo t = e2t 1 + t -t 1- t] " Example 2. Solve the initial value problem for (1) with -2 -1 0 A -1- -0 2 1 1 1 1 0 0 0 0 1 1 In this case, the matrix A has an eigenvalue A = 1 of multiplicity 4. , N is nilpotent of order 3. The solution of the initial value problem for (1) is therefore given by x(t) = et [I + Nt + N2t2/2]xo - e t 1 - t - t2/2 -2t - t2/2 -t - t2/2 -t - t2/2 t 1+t t t t2/2 t + t2/2 1 + t2/2 t2/2 0 0 0 1 xo.

8 of Chapter 2). In describing the topological behavior or qualitative structure of the solution set of a linear system, we do not distinguish between nodes and foci, but only if they are stable or unstable. There are eight different topological types of behavior that are possible for a linear system according to whether 6 76 0 and it has a source, a sink, a center or a saddle or whether 6 = 0 and it has one of the four types of behavior determined in Problem 4. Source Sink critical point Degenerate z 0 Saddle Figure 6.

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