Exercise 157

1 Find the particular integral of each of the following:

2 Find the y and the yc (and hence the general solution) of:

3 On the basis of the signs of the characteristic roots obtained in the preceding problem, analyze the dynamic stability of equilibrium. Then check your answer by the Routh theorem.

4 Without finding their characteristic roots, determine whether the following differential equations will give rise to convergent time paths:

(a) y"'(t) - 10y"(t) + 21y'(t) - 18>' = 3

(c) >•'"(/) + 4>'"(/) + 5y'(t) - 2y = -2

5 Deduce from the Routh theorem that, for the second-order linear differential equation y"(t) + axy'(t) + a2y = b, the solution path will be convergent regardless of initial conditions if and only if the coefficients a, and a2 are both positive.

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