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Math 54 - Quiz 08 Ans, Spring 2006

Math 54 - Quiz 08 Ans, Spring 2006 - The characteristic...

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Math 54, Quiz 8 Solutions Sections 202/203 GSI Carter 1. For each of the following statements, write the word “true” or “false.” (a) The initial value problem y = 2 3 y 3 , y (0) = 0 has a unique solution on the interval ( - 1 , 1). There is a mistake in this problem. (b) The initial value problem ty - 2 y + y = 0, y (1) = 1, y (1) = 0 has a unique solution on the interval (0 , ). True. See the theorems of 3.2. 2. Solve the following initial value problem: y - 2 y - 3 y = 0 y (0) = 1
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Unformatted text preview: The characteristic equation is given by 0 = r 2-2 r-3 = ( r-3)( r + 1). Therefore the general solution of the diﬀerential equation is y = c 1 e 3 t + c 2 e-t , and y = 3 c 1 e 3 t-c 2 e-t . Then the initial conditions give us 1 = c 1 + c 2 and 0 = 3 c 1-c 2 , so that c 1 = 1 4 and therefore c 2 = 3 4 . The solution to the initial value problem is then given by y = 1 4 e 3 t + 3 4 e-t . 1...
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