Exercise 157

1 Find the particular integral of each of the following: (a) y'"(0 + 2/'(0+/(0 + 2y = 8

(/>)/"(0+/'(') +3/(0 = 1 (c) 3y"'0 + 9/'(0 = 1 (</)/4)(/)+/'(0-4

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;

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 ax and a2 are both positive.

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