Consider the initial-value problem x 1 = a11x1 + a12x2 + b1(t), x 2 = a21x1 + a22x2 +
Chapter 10, Problem 31(choose chapter or problem)
Consider the initial-value problem x 1 = a11x1 + a12x2 + b1(t), x 2 = a21x1 + a22x2 + b2(t), x1(0) = 1, x2(0) = 2, where the ai j , 1, and 2 are constants. Show that the Laplace transforms of x1(t) and x2(t) must satisfy the linear system (s a11)X1(s) a12X2(s) = 1 + B1(s) a12X1(s) + (s a22)X2(s) = 2 + B2(s). This system can be solved quite easily (for example, by Cramers Rule) to determine X1(s) and X2(s), and then x1(t) and x2(t) can be obtained by taking the inverse Laplace transform. F
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