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In this problem we outline a different derivation of Eulers formula. a. Show that y1(t)
Chapter 3, Problem 20(choose chapter or problem)
In this problem we outline a different derivation of Eulers formula. a. Show that y1(t) = cos t and y2(t) = sin t are a fundamental set of solutions of y__+y = 0; that is, show that they are solutions and that their Wronskian is not zero. b. Show (formally) that y = eit is also a solution of y__+y = 0. Therefore, eit = c1 cos t + c2 sin t (31) for some constants c1 and c2. Why is this so? c. Set t = 0 in equation (31) to show that c1 = 1. d. Assuming that equation (15) is true, differentiate equation (31) and then set t = 0 to conclude that c2 = i. Use the values of c1 and c2 in equation (31) to arrive at Eulers formula.
Questions & Answers
QUESTION:
In this problem we outline a different derivation of Eulers formula. a. Show that y1(t) = cos t and y2(t) = sin t are a fundamental set of solutions of y__+y = 0; that is, show that they are solutions and that their Wronskian is not zero. b. Show (formally) that y = eit is also a solution of y__+y = 0. Therefore, eit = c1 cos t + c2 sin t (31) for some constants c1 and c2. Why is this so? c. Set t = 0 in equation (31) to show that c1 = 1. d. Assuming that equation (15) is true, differentiate equation (31) and then set t = 0 to conclude that c2 = i. Use the values of c1 and c2 in equation (31) to arrive at Eulers formula.
ANSWER:Step 1 of 10
The given differential equation is
First consider
Then