Let x1(t) and x2(t) be solutions of x' = x2 – t having initial conditions x1(0) = 0 and x2(0) = 3/4. Use Theorem to determine an upper bound for |x1(t) – x2(t)|, as long as the solutions x1(t) and x2(t) remain inside the rectangle defined by R – {(t, x): –1 ≤ t ≤ 1, –2 ≤ x ≤ 2}. Use your numerical solver to draw the solutions x1(t) and x2(t), restricted to the rectangular region R. Estimate maxR |x1(t) – x2(t)| and compare with the estimated upper bound.
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