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EXERCISE 9.2 1. Prove that if all the roots of the Characteristic Equation (9.17) with multiplicity greater than one have negative real parts and all simple roots have non-positive real parts, then all solutions of (9.16) are bounded.

In Chapter 2, the nth order linear homogeneous equation L(x) = x(") + apin-1). . + =0,--<I< (9.16) where a, ...., are real constants is studied. It is known that solutions of (9.16) exist on (-0,0). To determine a solution of (9.16), the characteristic equation is helpful. If the roots of the characteristic equation are known, the solution of (9.16) is completely determined. Thus, it is natural to expect that the behaviour of solutions of (9.16) depends on the nature of roots of the characteristic equation. Theorem 9.5 illustrates this point. Theorem 9.5 Let all the roots of the characteristic equation of (9.16) have negative real parts and let p>0 be such that -p> max Reely where 1 SSM ...,m are the distinct roots. Then there exists a constant K such that 1x(1)| SK exp(-pt), 120. Proof The characteristic equation of (9.16) is given by para + a, m-' +...+0,=0. (9.17) Let 2....,am bem distinct roots repeated 9....9 times respectively (9,+...+9m= n). Let 2;= a, + iß, j = 1,..., m. It is known that (refer to Chapter 2) x;(t) = * exp (2,0), OSP59; -1. Urhebe Dates Mar