# hw1 - a y t ˙ y t = sin 2 ωt ω is a real constant b ˙ y...

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ASE 330M Linear System Analysis Unique Number: 12495, Spring 2006 Homework #1 Math Review Complex Numbers and Ordinary Diﬀerential Equations Date given: January 24, 2006 Date Due: February 2, 2006 1. Convert the following complex numbers to polar coordinate (exponential) form: (a) - 2 + j 1 (b) 2 + j 4 (c) 1 - j 3 2. Convert the following complex numbers to rectangular (cartesian) form: (a) πe j 2 (b) 2 e - j 4 (c) - 3 e - j 3. Evaluate the expressions below and leave your results in rectangular form for z 1 = 2 + j 3, z 2 = 1 - j 3, and z 3 = j 2. (a) z 1 z 2 z 3 (b) ( z 1 + z 2 ) /z * 3 (c) ( z 1 /z 2 ) 3 4. Evaluate the expressions below and leave your results in rectangular form for z 1 = 0 . 5 e j 0 . 24 , z 2 = 4 e - j 1 . 2 , and z 3 = 3 e j 4 . (a) ( z 1 z 2 ) / ( z 2 + z 3 ) (b) ( z 1 + z * 2 + z 3 ) * 5. State whether the following ordinary diﬀerential equations are linear or nonlinear. You may assume t to be the independent variable and y ( t ) to be the dependent variable. Also determine the order of each of these diﬀerential equation systems: (i) ˙ y ( t ) + y ( t ) = 10 (ii) ¨ y ( t ) + ˙ y ( t ) y ( t ) = x ( t ) (iii) (1 - α y ( t ) + α ˙ y ( t ) = y 3 / 2 ( t ) for α = 1 (iv) ˙ y ( t ) x 2 ( t ) + 1 = y ( t ) + ¨ x ( t ) 6. Solve the following ﬁrst order ODEs:

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Unformatted text preview: a) y ( t ) ˙ y ( t ) = sin 2 ωt , ω is a real constant. b) ˙ y ( t ) = y ( t )tanh t You may leave the integration constants within the solution. 7. Solve the ﬁrst order linear ODE ˙ y ( t )-2 ty ( t ) =-2 t by the two diﬀerent methods: (i) separation of variables; and (ii) undetermined coeﬃcients. 1 8. Solve the following second linear ordinary diﬀerential equations: a) ¨ y ( t ) + ˙ y ( t )-2 y ( t ) = 0 subject to y (0) = 1 and y (1) = π . b) ¨ y ( t )-3 ˙ y ( t ) + 8 y ( t ) = 0 subject to y (0) =-1 and ˙ y (0) = 0. 9. Consider the following diﬀerential equation ¨ y ( t ) + 5 ˙ y ( t ) + 6 y ( t ) = ˙ x ( t ) + x ( t ) When the function x ( t ) = 6 t 2 is speciﬁed, use the method of undetermined coeﬃcients to ﬁnd the solution y ( t ) assuming initial conditions y (0) = 25 / 18, and ˙ y (0) =-2 / 3. 2...
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## This note was uploaded on 09/25/2011 for the course ASE 18510 taught by Professor Jeannefalcon during the Spring '10 term at University of Texas.

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hw1 - a y t ˙ y t = sin 2 ωt ω is a real constant b ˙ y...

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