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# midterm3 - dx dt =-2 x 3 y dy dt = 2 x-y d Consider the...

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Name: Section/Time of lecture: Professor/GSI: MIDTERM II Each part of a problem counts equally. To get full score you need to carefully explain what you did. No calculators allowed. 1

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Problem Points Score 1 60 2 60 3 60 4 60 + 10 10 TOTAL 250 2
Problem 1. a) Find a particular solution of y + y + y = e 2 x . b) Find a particular solution of y - 4 y = e 2 x . c) Use the method of variation of parameters to find a particular solution of y - y = xe 2 x d) A mass of 10 kg is attached to a spring which stretches 1m if you use a force of 5N. A force F 0 = (sin( ωt )) N is applied to the mass. For which value of ω will resonance occur? [There is no need to change units. There is no penalty for messing up on units.] 3

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Problem 2. a) Transform the system x = 2 x - y y = 2 y - x into an equivalent system of first order differential equations: x 1 = x 2 = x 3 = x 4 = b) Solve dx dt = 2 y, x (0) = 1 dy dt = - 3 x, y (0) = 2 4
c) Use the method of elimination to find the general solution of

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Unformatted text preview: dx dt =-2 x + 3 y, dy dt = 2 x-y d) Consider the system: x = 2 x-y y = 3 x + y Estimate x (2) , y (2) using Eulers method with stepsize h = 1 if x (0) = 1 , y (0) = 2 . 5 Problem 3. a) Calculate the matrix products AB and BA. A = ± 1 + i 2 i ² B = ± i 2 0 3 ² b) Calculate the determinant of ± 3 + i-i 6 2 ² 6 c) Find the eigenvalues and eigenvectors of ±-1 2-9 5 ² d) Use the Eigenvalue Method to ﬁnd the general solution of dx dt =-x + 13 y, dy dt =-2 x + y 7 Problem 4. a) Find the general solution of x 00 =-3 x + y y 00 = 2 x-2 y b) dx dt = 3 x, dy dt =-2 y Draw a phase portrait and Direction ﬁeld. 8 c) dx dt =-5 y, dy dt =-2 x Decide if (0 , 0) is stable. d) dx dt = x + x 2-2 xy, dy dt = y + 2 y 2-3 xy Find all critical points. 9...
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midterm3 - dx dt =-2 x 3 y dy dt = 2 x-y d Consider the...

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