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Math · Calculus
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6. Answer the quest ns below for the following system: 5 -3 of the linear system. (ii) What type of equilibrium is the origin Center, spiral sink, or spiral source? (iii) What is the natural frequency of the oscillations in the system? What is the natural period length? (iv) If Y(0) (1,1 then what is the derivative Y(0)? Does the system spiral clockwise or counter-clockwise? (v) Find two linearly independent solutions Ye(t) and Y such that the general solution of the system is of the form Y(t) kiY (t) k2Ym t). (vi) Find the particular solution given by the initial condition Y(0) (1,1). (vii) Using the formula you derived in part (vi), plot the z() and y() curves on the same graph for 0 st S 3. Make sure to indicate which curve is z(t and which curve is y(t). (Hint: You may need to use the symbol to perform element-trise multiplication of vectors. For erample, to plot the function z(t) e sin(4), you could use the code below) t min 0; t-max 3; t-step 01 time t-min t-step:t.nax; x exp(-2. time). sinC4 time); plot (time,x) (viii) On the z x y plane, plot the direction field of our linear system. On top of your direction field, plot the solution curve Y(t) (z(t),y(t)) you found in part (vi). The dimensions of your graph should be I-2,2 x (-2,2 and you should plot the curve Y() for (Hint: If your solution curve does not run parallel with the direction field, then you have made a mistake somewhere.) If Y(0)=(1,1) what is the derivative Y'(0)? Find two linearly independent solutions Yre(t) and Yim(t) such that the general solution of the system is of the form Y (t) = k1Yre(t) + k2Yim(t). Find the particular solution given by the initial condition Y (0) = (1, 1).

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