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Unformatted text preview: MthSc 208, Spring 2011 (Differential Equations) Dr. Macauley HW 7 Due Friday February 18th, 2011 (1) Sove the following initial value problems. (a) y - y - 2y = 0, y(0) = -1, y (0) = 2 (b) y - 4y - 5y = 0, y(1) = -1, y (1) = -1 (c) y + 25y = 3, y(0) = 1, y (0) = -1 (d) y - 2y + 17y = 0, y(0) = -2, y (0) = 3 (2) Find the general solution to the following 2nd order linear inhomogeneous ODEs, by solving the associated homogeneous equation, and then finding a constant (particular) solution. (a) y + y - 12y = 24 (b) y = -4y + 3 (3) As we've seen, to solve ODE of the form y + py + qy = 0 , p and q constants
rt we assume that the solution has the form e , and then we plug this back into the ODE to get the characteristic equation: r2 +pr +q = 0. Given that this equation has a double root r = r1 (i.e., the roots are r1 = r2 ), show by direct substitution that y = tert is a solution of the ODE, and then write down the general solution. [Hint: If there's a double-root, then it must be - p . Why?] 2 (4) Suppose that z(t) = x(t) + iy(t) is a solution of z + pz + qz = Aeit . Substitute z(t) into this equation above. Then compare (equate) the real and imaginary parts of each side to prove two facts: x + px + qx = A cos t y + py + qy = A sin t . Write a sentence or two summarizing the significance of this result. (5) Solve the following initial value problems using the method of undetermined coefficients. (a) y + 3y + 2y = -3e-4t , y(0) = 1, y (0) = 0 (b) y + 2y + 2y = 2 cos 2t, y(0) = -2, y (0) = 0 (c) y + 4y + 4y = 4 - t, y(0) = -1, y (0) = 0 (d) y - 2y + y = t3 , y(0) = 1, y (0) = 0 (6) Solve the following first order differential equations using the method of undetermined coefficients. (a) y - 3y = 5e2t (b) y + 2y = 3t (c) T = k(-t - T ) (d) T = k(A sin t - T ) ...
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