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\begin{document}

\title{Applied Differential Equations and Modeling}
\author{Homework 10}
\date{Due in class Tuesday, April 30, 2019}
\maketitle

\begin{enumerate}

\item Find the Laplace transform of the given function.
\begin{enumerate}
\item $\displaystyle f(t) = t^{10}$
\item $\displaystyle f(t) = \e^{2t} \, \cos 3 t$
\item $\displaystyle f(t) =
\begin{cases}
t & \text{for } 0 \leq t \leq 1 \\
1 & \text{for } t > 1
\end{cases}
$
\item $\displaystyle f(t) = t^n \, \e^{a t}$
\end{enumerate}

\item Show that the Laplace transform $\mathcal L$ satisfies
\[
  \mathcal L \int_0^t f(\tau) \, \d \tau
  = \frac1s \, \mathcal L(f)
\]
assuming that the transforms on the left and on the right hand sides
are well defined.

\item Apply the Laplace transform to the  given initial value
problem, and solve the resulting algebraic expression for the Laplace
transform of the solution.  (Note: the inverse transform, which is
necessary to find the solution itself, will be discussed next week, it
is not required here.)

\begin{enumerate}
\item $\displaystyle y'' - y' - 6y = 0$

with $y(0)=1$, $y'(0)=-1$

\item $\displaystyle y'' + \omega^2 \, y = \cos 2t$

for $\omega^2 \neq 4$ with $y(0)=1$, $y'(0)=0$


\item $\displaystyle y''''- 4y = 0$

with $y(0)=1$, $y'(0)=0$, $y''(0)=-2$, $y'''(0)=0$

\end{enumerate}


\item For a function $f(t)$, write $F(s)$ to denote the Laplace
transform.  Prove the following.
\begin{enumerate}
\item $\displaystyle \mathcal L \biggl( \frac{f(t)}t \biggr) = \int_s^\infty F(u) \,
\d u$
%\item $\displaystyle \mathcal L (\e^{ct} \, f(t)) =  F (s-c)$
\item $\displaystyle \mathcal L (f(ct)) = \frac1c \, F \Bigl(
\frac{s}c \Bigr)$
\end{enumerate}


\end{enumerate}
\end{document}
