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s11_mthsc208_hw08

s11_mthsc208_hw08 - MthSc 208 Fall 2010(Differential...

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MthSc 208, Fall 2010 (Differential Equations) Dr. Matthew Macauley HW 8 Due Tuesday September 21st, 2010 (1) If y f ( t ) is a solution of y 00 + py 0 + qy = f ( t ) and y g ( t ) is a solution of y 00 + py 0 + qy = g ( t ) , show that z ( t ) = αy f ( t ) + βy g ( t ) is a solution of y 00 + py 0 + qy = αf ( t ) + βg ( t ) , where α and β are any real numbers, by plugging it into the ODE. (2) Find the general solution to the following 2 nd order linear inhomogeneous ODEs. (a) y 00 + 2 y 0 + 2 y = 2 + cos 2 t (b) y 00 + 25 y = 2 + 3 t + 4 cos 2 t (c) y 00 - y = t - e - t . (3) (a) Find the general solution of y 00 + 3 y 0 + 2 y = te - 4 t . (Look for a particular solution of the form y p = ( at + b ) e - 4 t .) (b) Use a similar approach as above to find a solution to the differential equation y 00 + 2 y 0 + y = t 2 e - 2 t . (4) Find the general solution of y 00 + 2 y 0 + 2 y = e - 2 t sin t . (Look for a particular solution of the form y p =
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