s11_mthsc208_hw08

s11_mthsc208_hw08 - MthSc 208, Fall 2010 (Differential...

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Unformatted text preview: MthSc 208, Fall 2010 (Differential Equations) Dr. Matthew Macauley HW 8 Due Tuesday September 21st, 2010 (1) If yf (t) is a solution of y + py + qy = f (t) and yg (t) is a solution of y + py + qy = g(t) , show that z(t) = yf (t) + yg (t) is a solution of y + py + qy = f (t) + g(t) , where and are any real numbers, by plugging it into the ODE. (2) Find the general solution to the following 2nd order linear inhomogeneous ODEs. (a) y + 2y + 2y = 2 + cos 2t (b) y + 25y = 2 + 3t + 4 cos 2t (c) y - y = t - e-t . (3) (a) Find the general solution of y + 3y + 2y = te-4t . (Look for a particular solution of the form yp = (at + b)e-4t .) (b) Use a similar approach as above to find a solution to the differential equation y + 2y + y = t2 e-2t . (4) Find the general solution of y + 2y + 2y = e-2t sin t. (Look for a particular solution of the form yp = e-2t (a cos t + b sin t).) (5) For the following exercises, rewrite the given function in the form y = A cos(t - ) = A cos t - and then plot the graph of this function. (a) y = cos 2t + sin 2t (b) y = cos t - sin t (c) y = cos 4t + 3 sin 4t (d) y = - 3 cos 2t + sin 2t. (6) Consider the undamped oscillator mx + kx = 0, x(0) = x0 , x (0) = v0 . (a) Write the particular solution of this initial value problem in the form x(t) = a cos t+ b sin t (i.e., determine a, b, and .). (b) Write your solution in the form x(t) = A cos(t - ) (i.e., determine A). , ...
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This note was uploaded on 03/11/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.

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