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
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Problem 1. a) Find a particular solution of y 00 + y 0 + y = e 2 x . b) Find a particular solution of y 00 - 4 y = e 2 x . c) Use the method of variation of parameters to find a particular solution of y 00 - 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 00 = 2 x 0 - y y 00 = 2 y 0 - x into an equivalent system of first order differential equations: x 0 1 = x 0 2 = x 0 3 = x 0 4 = b) Solve dx dt = 2 y, x (0) = 1 dy dt = - 3 x, y (0) = 2 4
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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 B = i 2 0 3 b) Calculate the determinant of 3 + i-i 6 2 6 c) Find the eigenvalues and eigenvectors of -1 2-9 5 d) Use the Eigenvalue Method to nd 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 eld. 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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