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MATH150T7 - MATH 150 Resonance Cheung Man Wai Tutorial 7 1...

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MATH 150 Resonance Cheung Man Wai Tutorial 7 1 When inhomogeneous term is a solution of the homogeneous solution Notice that in the examples in our previous tutorial, we have g ( t ) = x h ( t ). What will happen if g ( t ) = x h ( t )? The strategy is to multiply t to your guessed x p ( t ). Example 1 Find a particular solution of ¨ x - 3 ˙ x - 4 x = 2 e - t . Remark . Proceeding as in the lecture note, we assume that x ( t ) = Ae - t . By substituting in the question, we then obtain ( A + 3 A - 4 A ) e - t = 2 e - t . Since the left-hand-side of above vanishes, there is no choice of A that satisfies this equation. Thus there is no particular solution of the question of the assumed form. The reason for this possibly unexpected result becomes clear if we solve the homogeneous equation ¨ x - 3 ˙ x - 4 x = 0 that corresponds to the question. A fundamental set of solution of homogeneous part is x 1 ( t ) = e - t and x 2 = e 4 t . Thus our assumed particular solution is actu- ally a solution of the homogeneous equation. Consequently, it cannot possibly be a solution of the inhomogeneous equation. To find a particular solution, we
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