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# sol3 - MATHEMATICS 3161 Fall 2009(2009.9 2009.12 Assignment...

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Unformatted text preview: MATHEMATICS 3161: Fall 2009 (2009.9 - 2009.12) Assignment # 3 (Due Date: Oct. 19) 1. Transform the given equation or initial value problem into a system of first order equations or a system of first order equations with initial conditions. a) u (5) + tu (3) + ( t − 1) u = 0, b) u ′′ + 0 . 25 u ′ + 4 u = 2 cos 3 t, u (0) = 1 , u ′ (0) = − 2 Solution : Let x 1 = u, x 2 = u ′ , x 3 = u ′′ , x 4 = u ′′′ , x 5 = u (4) , then a) the system is x ′ 1 = x 2 , x ′ 2 = x 3 , x ′ 3 = x 4 , x ′ 4 = x 5 , x ′ 5 = − tx 4 − ( t − 1) x 1 b) the system is x ′ 1 = x 2 , x ′ 2 = − . 25 x 2 − 4 x 1 + 2 cos 3 t with initial condition x 1 (0) = 1 , x 2 (0) = − 2 2. If A = parenleftBigg 1 + i − 1 + 2 i 3 + 2 i 2 − i parenrightBigg and B = parenleftBigg i 3 2 − 2 i parenrightBigg , find a ) A − 2 B, b ) AB, c ) BA Solution : a) A − 2 B = parenleftBigg 1 − i − 7 + 2 i − 1 + 2 i 2 + 3 i parenrightBigg b) AB = parenleftBigg − 3 + 5 i 7 + 5 i 2 + i 7 + 2 i parenrightBigg c ) BA = parenleftBigg 8 + 7 i 4 − 4 i 6 − 4 i − 4 parenrightBigg 3. If x = 2 3 i 1 − i and y = − 1 + i 2 3 − i , find a ) x T y , b ) y T y , c ) ( x , y ) , d ) ( y , y ) Solution : a) x T y = 2( − 1 + i ) + 6 i + (1 − i )(3 − i ) = 4 i b) y T y = ( − 1 + i ) 2 + 4 + (3...
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sol3 - MATHEMATICS 3161 Fall 2009(2009.9 2009.12 Assignment...

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