# m238sum08t2 - of a particular solution to the related...

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name : Mathematics 238 test two Tuesday, July 22, 2008 please show your work to get full credit for each problem 1–7 . Find the characteristic equation & the general (real-valued) solution of each of the following linear homogeneous differential equations. 1 . y 00 + 25 y 0 + 100 y = 0 2 . y 00 + 20 y 0 + 100 y = 0 3 . y 00 + 12 y 0 + 100 y = 0 4 . y 00 + 100 y = 0 5 . y (4) + 100 y (2) = 0 6 . ( D + 6)( D - 3) 2 ( D 2 + 5) y = 0 7 . ( D 2 + 100) 2 y = 0 8 . Describe the behavior of the solutions to problem ( 2 ) as t → ∞ . 9 . Describe the behavior of the solutions to problem ( 4 ) as t → ∞ . 10-11 . Find real-valued general solutions to each of the following degree two Euler differ- ential equations. 10 . t 2 y 00 - 3 ty 0 - 5 y = 0 11 . t 2 y 00 + ty 0 + y = 0 12 . Rewrite 6 cos 3 t - 5 sin 3 t in the format A cos(3 t - δ ). 13 . Use Euler’s formula to calculate the real part of (3 - 4 i ) e ( - 2+5 i ) t 14 . Solve for R and θ : 3+3 i = Re 15 . Solve the initial value problem: y 00 + 9 y = 0 y (0) = 6 y 0 (0) = - 15

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page two 16a . Find the solution to the homogeneous equation
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Unformatted text preview: of a particular solution to the related non-homogeneous equation y 00 + 4 y + 3 y = 2 + 4 t + cos 3 t + 2 te t + 5 e-t + 6 te-3 t It is not necessary to solve for the value of the coeﬃcients. 17 . Solve the initial value problem: y 00 + 2 y + y = 4 sin t y (0) = 0 y (0) = 1 18 . Use the method of variation of parameters to ﬁnd the general solution of y 00 + 4 y = sec 2 t The solution has the format u 1 y 1 + u 2 y 2 , where y 1 and y 2 are homogeneous solutions, g ( t ) is the right-hand side function, u 1 = ± ± ± ± ± y 2 g y 2 ± ± ± ± ± ± ± ± ± ± y 1 y 2 y 1 y 2 ± ± ± ± ± and u 2 = ± ± ± ± ± y 1 y 1 g ± ± ± ± ± ± ± ± ± ± y 1 y 2 y 1 y 2 ± ± ± ± ±...
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