m2420-ass11s-F2019.pdf - Math 2420(Fall 2019 Assignment#11 Problem#1 Use Laplace transforms to solve the following IVPs t 00 y 4y = 1(a y(0 = 0 y 0(0 =

# m2420-ass11s-F2019.pdf - Math 2420(Fall 2019 Assignment#11...

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Math 2420 (Fall 2019) Assignment #11 Problem #1: Use Laplace transforms to solve the following IVPs: (a) y 00 + 4 y = t if 0 t 1 . 1 if 1 t y (0) = 0 , y 0 (0) = 0 . (b) y 00 + 5 y 0 + 6 y = g ( t ) y (0) = 0 , y 0 (0) = 2 . , g ( t ) = 0 if 0 t < 1 , t if 1 t < 5; 1 if 5 t. (c) y 00 + 3 y 0 + 2 y = f ( t ) y (0) = 2 , y 0 (0) = - 1 . f ( t ) = e - t if 0 t 3 , 1 if 3 < t. (d) y 00 + 9 y = g ( t ) y (0) = 1 , y 0 (0) = 3 g ( t ) = sin 3 t if t < π 6 , 1 + sin(3 t - π 2 ) if π 6 t. Solution: 1
(a). This equation can be written as y 00 + 4 y = t + u ( t - 1)(1 - t ) y (0) = 0 , y 0 (0) = 0 , thus we have Y ( s ) = L { t - u ( t - 1)( t - 1) } s 2 + 4 = 1 s 2 - e - s 1 s 2 s 2 + 4 = 1 - e - s s 2 ( s 2 + 4) = 1 4 · 1 s 2 - 1 4 · 1 s 2 + 4 (1 - e - s ) . Therefore, we obtain the solution y ( t ) = L - 1 { Y ( s ) } = L - 1 1 4 · 1 s 2 - 1 8 · 2 s 2 + 4 (1 - e - s ) = 1 4 sin t - 1 8 sin 2 t - u ( t - 1) 1 4 sin( t - 1) - 1 8 sin 2( t - 1) .

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