Question

I dont know how to incorporate the frequency from the start. I need help!!! Please let me know what to do

1 - One engineer is studying a specific mathematical modeling for one of his projects and came up with the following system of differential equations: But the system above does not produce an oscillatory behavior as he expected. (a) Solve the system (1) to make sure the engineer did not make any mistake. (b) After realizing the system did not produce the desired output, he decided to study the effect of one of its components and proposed to himself the following change: y' = -10x + 8y, where ε is a real number. He wants to check if a perturbation in one of the parameters will produce the desired output. He asked for help for the Math 2240 students. For which value of ε the system would produce an oscillatory behavior with frequency 100 Hz? (Hint: This is the condition on the imaginary part of the complex eigenvalues of the coefficient matrix.) (c) The students of CPP after solving item (b) realized that he also needs to the trace of the coeffi consider some changes in one of the coefficients of cient matrix since the results from item (b) produced an unstable behavior (Real part of the complex eigenvalues is positive). Then, they proposed the following system to study: -10x + (84 δ)y, y' = Show that in order to have an oscillatory behavior at 100 Hz, δ and ε are related by the following equation: 39984 28+ (1+ 6)2 40 e(6) What is the optimal value of δ such that it produces the minimum value of ε? Does the optimum value of δ produces a complex eigenvalue with negative real part? Cornpute ε for the optimal value of δ and compare with the value found in item (b) (d) Write the general solution of system (3) considering the values of e and 6 found in item (c) (e) Present your conclusions and a plot of the curve (4) to support your find- ings

Answer 1

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