m238sp05t2 - Differential Equations test two Monday, May...

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Unformatted text preview: Differential Equations test two Monday, May 16, 2005 please show your work in order to get full credit 1. For which values of t is the solution to the following differential equation guaranteed to exist? (t2 − 9)y + (cos t)y + y = 2t and y (0) = −4 2. Calculate the Wronskian of the functions f (t) = e2t and g (t) = e−t . 3. Which of the following are linearly independent sets of functions? (a) {f (t) = t, g (t) = t2 } t (b) {f (t) = cos t, g (t) = t cos t} (c) {f (t) = e − sin t, g (t) = sin t − et } 4. Find the real part of the complex-valued function f (t) = (2 + 3i)e(−1+4i)t . 5. Find the real-valued general solution of each differential equation: (a) y − 2y − 8y = 0 (b) y + 4y + 40y = 0 (d) y + 2y = 12 − 4 cos 2t (c) D(D − 2)(D + 4)2 (D2 + 4)y = 0 6. Solve the initial value problem: y − 2y − 8y = 0 and y (0) = 1 and y (0) = 0. 7. Given the differential equation y − 2y − 8y = 9 + 18et • find a particular solution of this differential equation. • find the general solution of this equation. • find a solution which satisfies initial conditions y (0) = 0 and y (0) = 1. 8. Use the variation of parameters technique to solve the differential equation y + 16y = sec 4t (use the formulas that (u1 ) = −y2 (t)g (t)/W and (u2 ) = y1 (t)g (t)/W , where W is the Wronskian of homogeneous solutions y1 (t) and y2 (t)) 9. Suppose that y1 (t) and y2 (t) are solutions to the linear equation y + p(t)y + q (t)y = 0, and that the Wronskian of the two functions is not equal to zero. Based on this information, decide which of the following statement(s) are true: • {y1 (t), y2 (t)} is a linearly dependent set of functions. • y = c1 y1 (t) + c2 y2 (t) is a general solution to the differential equation. ...
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This note was uploaded on 11/01/2010 for the course ENGINEERIN PHYS222 taught by Professor Ewqfqw during the Spring '09 term at University of Washington.

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