= u” (t) + Au() + Ba(t) = f(t), t€ IR, where A, B are real numbers and f:R → R is a continuous function. Let ri, r2 be t

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U T Au Ba T F T T Ir Where A B Are Real Numbers And F R R Is A Continuous Function Let Ri R2 Be T 1 (147.51 KiB) Viewed 41 times

= u” (t) + Au() + Ba(t) = f(t), t€ IR, where A, B are real numbers and f:R → R is a continuous function. Let ri, r2 be the solutions of its characteristic equation: p2 + Ar +B=0. (a) Prove that, if rı r2, then the general solution to equation (2.42) is: eri(t-s) – cr2(t-s). u(t) = -) f(s)ds + Cie"it + Czerzt 11-12 where C and C2 are constants. (6) Prove that, if ri = r2, then the general solution to equation (2.42) is: == ſ e u(t) = = = f 0 )1s+ 1 (t – she'l(t-s) f(s)ds + (C1 + C2t)e"it, where C and C2 are constants. Hint: Use the “variation of constants” method and the formula: | s()dt = [*s(5)ds +C; where CER, which is true for all continuous functions f:R → R.