Exercise 10

Let $y = y(t)$ and $y'(t)$ be piecewise continuous, exponentially bounded functions and assume that $y''$ is exponentially bounded.

1. Prove that $${\mathcal L}(y'')(s) = s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0)$$ for large values of $s$, where $Y(s)$ is the Laplace transform of $y$.

2. If $y = y(t)$ and all of its derivatives up to order $k - 1$ are piecewise continuous, exponentially bounded functions and $y^{(k)}$ is piecewise continuous, prove that $$\begin{gathered} {\mathcal L}(y^{(k)})(s) = s^k {\mathcal L}(y)(s) - s^{k - 1}y(0) - \cdots - s y^{(k -2)}(0) - y^{(k -1)}(0) \\ = s^k Y(s) - s^{k - 1}y(0) - s^{k - 1}y(0)- \cdots - s y^{(k -2)}(0) - y^{(k -1)}(0). \end{gathered}$$