lecture_18 (dragged) 3 - 4 MA 36600 LECTURE NOTES: FRIDAY,...

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Unformatted text preview: 4 MA 36600 LECTURE NOTES: FRIDAY, FEBRUARY 27 must be the function ￿ ￿ 1 y (t) = c1 y1 (t) + c2 y2 (t) + Y (t) = c1 e−t + c2 e4t + − e2t 2 for some constants c1 and c2 . Example. Consider the differential equation y ￿￿ − 3 y ￿ − 4 y = 2 sin t. Again, we will find the general solution of this equation. To do so, we compute functions y1 = y1 (t), y2 = y2 (t), and Y = Y (t). First we find the functions y1 and y2 as solutions to the homogeneous equation y ￿￿ − 3 y ￿ − 4 y = 0. We found above that a fundamental set of solutions is given by y1 (t) = e−t and y2 (t) = e4t . Second we seek a function Y = Y (t) as a solution to the nonhomogeneous equation Y ￿￿ − 3 Y ￿ − 4 Y = 2 sin t. It suffices just to find one solution Y = Y (t). To this end, we make a guess that such a solution is in the form Y (t) = A cos β t + B sin β t for some constants A, B , and β . We have the derivatives Y Y￿ Y ￿￿ This gives the expression = A cos β t + B sin β t = β B cos β t + −β A sin β t = −β 2 A cos β t + −β 2 B sin β t ￿ ￿ ￿ ￿ Y ￿￿ − 3 Y ￿ − 4 Y = −β 2 A − 3 β B − 4 A cos β t + −β 2 B + 3 β A − 4 B sin β t = 2 sin t. We now choose β = 1, so that ￿ ￿ ￿ ￿ 2 sin t = Y − 3 Y − 4 Y = −5 A − 3 B cos t + 3 A − 5 B sin t. ￿￿ ￿ It makes sense to choose A and B such that −5 A − 3 B = 0 3A − 5B = 2 We can solve for A and B by multiplying the first equation by −5 and the second by 3; or multiplying the first equation by 3 and the second by 5: 25 A + 15 B 9 A + −15 B 34 A =0 =6 =6 −15 A + −9 B 15 A + −25 B −34 B =0 = 10 = 10 This gives A = 3/17 and B = −5/17, so that one solution to the differential equation is 3 5 Y (t) = cos t − sin t. 17 17 Hence the general solution to the nonhomogeneous differential equation must be the function y ￿￿ − 3 y ￿ − 4 y = 2 sin t y (t) = c1 y1 (t) + c2 y2 (t) + Y (t) = c1 e−t + c2 e4t + for some constants c1 and c2 . ￿ 3 5 cos t − sin t 17 17 ￿ ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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