In this problem we outline a different derivation of Eulers formula. a. Show that y1(t)

Chapter 3, Problem 20

(choose chapter or problem)

Get Unlimited 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.

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

Add to cart


Study Tools You Might Need

Not The Solution You Need? Search for Your Answer Here:

×

Login

Login or Sign up for access to all of our study tools and educational content!

Forgot password?
Register Now

×

Register

Sign up for access to all content on our site!

Or login if you already have an account

×

Reset password

If you have an active account we’ll send you an e-mail for password recovery

Or login if you have your password back