Fall 2006 - Hall's Class - Practice Exam 2

Fall 2006 - Hall's Class - Practice Exam 2 - Math 20D...

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Unformatted text preview: Math 20D Practice Exam II Solutions 1. (a) (5 points) Find the general solution to the homogeneous differ- ential equation y 00 + 4 y + 4 y = 0 . y = c 1 e- 2 t + c 2 te- 2 t (b) (8 points) Use the method of Undetermined Coefficients to find a particular solution to the non-homogeneous differential equation y 00 + 4 y + 4 y = 2 t + 1 . Guess Y = at + b . Plug in to get 4 a + 4 at + 4 b = 2 t + 1, i.e., a = 1 / 2 ,b =- 1 / 4 Y = (1 / 2) t- 1 / 4 (c) (8 points) Use the method of Variation of Parameters to find a particular solution to the non-homogeneous differential equation y 00 + 4 y + 4 y = e- 2 t 1- t 2 . (Hint: R 1 1- x 2 dx = arcsin( x ) + C ) W ( y 1 ,y 2 ) = e- 2 t te- 2 t- 2 e- 2 t e- 2 t- 2 te- 2 t = e- 4 t u 1 =- y 2 g ( t ) W ( y 1 ,y 2 ) =- t 1- t 2 u 1 = 1- t 2 u 2 = y 1 g ( t ) W ( y 1 ,y 2 ) = 1 1- t 2 u 2 = sin- 1 t Y = 1- t 2 e- 2 t + (sin- 1 t ) te- 2 t (d) (4 points) Write the general solution to the the non-homogeneous differential equation y 00 + 4 y + 4 y = 2 t + 1 + e- 2 t 1- t 2 . y = c 1 e- 2 t + c 2 te- 2 t + (1 / 2) t- 1 / 4 + 1- t 2 e- 2 t + (sin- 1 t ) te- 2 t 2. Consider the initial value problem 2 t 2 y 00 + ty- 3 y = 0 ,y (1) =- 2 ,y (1) = 1 . 1 (a) (5 points) Without solving the initial value problem, explain why its solution will only be defined for t > 0....
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Fall 2006 - Hall's Class - Practice Exam 2 - Math 20D...

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