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Consider the linear equation y' + by = f(t). Suppose that b > 0 is constant,...

Consider the linear equation y' + by = f(t). Suppose that b > 0 is constant, and |f| is bounded by some M > 0 (Namely, |f(t)| < M for every real number t).

Show that if y(t) is a solution of the equation then there is a constant C such that |y(t)| ≤ C + (M/ b) for all t ≥ 0.

(Hint: Use the fact µ(t) = ebt is an integrating factor, and that | ∫f(t) · µ(t)dt | ≤ ∫ |f(t) · µ(t)| dt).

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