m2420-ass7s-2019.pdf - Math 2420(Fall 2019 Assignment#7 Problem#1 Use the method of variation of parameters to solve the following equations(a y 00 2y 0

# m2420-ass7s-2019.pdf - Math 2420(Fall 2019 Assignment#7...

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Math 2420 (Fall 2019) Assignment #7 Problem #1: Use the method of variation of parameters to solve the fol- lowing equations (a) y 00 + 2 y 0 + 2 y = 1 e t sin t ; (b) y 00 + y = 1 t 2 ; (c) ty 00 - y 0 - 4 t 3 y = 16 t 3 e t 2 , knowing that (check it) that y 1 = e t 2 and y 2 = e - t 2 , are solutions of the homogeneous equations. (d) y 00 - 2 tan t y 0 = 1 Solution: (a). r 2 + 2 r + 2 = 0 ( r + 1) 2 = - 1 r = - 1 ± i Fundamental solution set: { e - t cos t, e - t sin t } So y p = u 1 y 1 + u 2 y 2 and the augmented matrix in this case has the form: " e - t cos t e - t sin t 0 e - t ( - cos t - sin t ) e - t (cos t - sin t ) 1 e t sin t # So that W [ y 1 , y 2 ] = e - 2 t cos t (cos t - sin t ) - e - 2 t sin t ( - cos t - sin t ) = e - 2 t 1
thus u 0 1 = - e - t sin t e t sin te - 2 t = - 1 u 1 = - Z dt = - t and u 0 2 = e - t cos t e t sin te - 2 t = sin t cos t = tan t u 2 = Z sin t cos t dt = ln | sin t | Hence y p = - te - t cos t + ln | sin t | e - t sin t and we obtain the general solution: y = C 1 e - t cos t + C 2 e - t sin t - te - t cos t + ln | sin t | e - t sin t or y = e - t [( C 1 - t ) cos t + ( C 2 + ln | sin t | ) sin t ] (b).

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