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Stability of Euler’s method Consider the initial value problem y'(t) = –ay, y(0) = 1, where a > 0; it has the exact solution y(t) = e–at, which is a decreasing function.
a. Show that Euler’s method applied to this problem with time step h can be written u0 = 1, uk+l = (1 – ah)uk, for k = 0, 1, 2,….
b. Show by substitution that uk = (1 – ah)k is a solution of the equations in part (a), for k = 0,1,2, ….
c. Explain why as k increases the Euler approximations uk = (1 – ah)k decrease in magnitude only if |1 – ah| < 1.
d. Show that the inequality in part (c) implies that the time step must satisfy the condition
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