Please explain why the roots that were used were. Working Backwards Write the st

Question

Please explain why the roots that were used were.

Working Backwards Write the standard form (equation (16) with leading coefficient an = 1) of the nth order linear homogeneous differential equation with real coefficients whose roots are given in Problem 3rd order, two of the roots are r = -2 + i, 2 + i A Given two of the roots are r = -2 + i, 2 + i The characteristic equation that contains three roots where two of the roots are equal to the following written thereafter, r_1, 2 = 2 plusminus i (r - 2)(2 + i)(2 - i) = 0 (r - 2)(r^2 - 4r + 5) = 0 r^3 - 6r^2 + 13r - 10 = 0 Therefore, the 3^rd order differential equation that represents this characteristic equation is given as follows, y"' - 6y" + 13y' - 10y = 0

Explanation / Answer

Question is not proper and the step-by-step solution provided is not correct. The equation has to be a minimum of fourth order. If the characteristic polynomial has real coefficients and has 2+i and -2+i as roots, then their complex conjugates should also be roots of the characteristic polynomial. Hence the degree of the characteristic polynomial should be minimum four and thus the differential equation should be of minimum fourth order.

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