Math homework

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Math 242 HW 8 (Section 3.4) Be sure to carefully justify all of your answers! Section 3.4.1 Method of undetermined coefficients 1. Find the general solution to each differential equation: (a) x00 − 2x0 = tet (b) x00 − 2x0 = −12t2 + 16t − 8 (c) x00 − 2x0 = 6e2t − 10 sin t (a) x(t) = c1 + c2 e2t + −tet   (b) x(t) = c1 + c2 e2t + t(2t2 − t + 3)   (c) x(t) = c1 + c2 e2t + t(3e2t ) + (2sin t − 4cos t) 2. Determine the form of a particular solution to each of the following differential equations: (a) x00 − 6x0 + 9x = t cos (2t) + (t − 2)e3t (b) x00 + 2x0 + 2x = 3e−t + 2e−t cos t + 4t2 sin t (c) x00 − 4x0 + 4x = 2t2 + 3te2t + sin 2t (d) x00 + 4x = 4t cos (2t) 1 Section 3.4.2 Variation of parameters 1. Find the general solution to each of the following differential equations: (a) x00 + 4x = 4 sec (2t) (b) t2 x00 − tx0 + x = t (c) t2 x00 − 4tx0 + 6x = t2 ln(t) (a) x = c1 cos(2t) + c2 sin(2t) + [(ln |cos(2t)|) (cos(2t)) + (2t) (sin(2t))] h  i (b) x = c1 t + c2 t lnt + − 12 (ln t)2 (t) + (ln t) (tln t) h    i (c) x = c1 t2 + c2 t3 + − 21 (ln t)2 t2 + − 1t ln t − 1t t3 2
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