\documentclass[11pt]{amsart}
\setlength{\textheight}{10in}
\setlength{\topmargin}{-1in}
\usepackage{graphicx}
\pagestyle{empty}
\newcommand{\mat}[1]{\begin{bmatrix}#1\end{bmatrix}}
\newcommand{\dis}{\displaystyle}
\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Math 305 \hfill Homework Set 6  \hfill Spring 2017

\begin{center}Due Monday February 27\end{center}

{\bf I.} For each differential equation below, find the general 
solution.


\begin{enumerate}
\item $9y'' + 6 y' + y = x^2$

\item $9y'' + 6y' + y = e^{-\frac{x}{3}}$

\item $y'' + 3y' - 4y = 2x + e^x$

\item $y'' + 3y' - 4y = \sin x$

\item $y'' + 4 y = x$

\item $y''  + 4 y = \cos x $

\item $y''  +4 y = \cos 2x $

\end{enumerate}


{\bf II.} Determine if each pair of functions is linearly dependent
or independent on the given interval.

\begin{enumerate}
\item $\{ x^2 + x, x \}$ on $(-\infty,\infty)$

\item $\{ \sin x, \sin 2x \}$ on $(-\infty,\infty)$

\item $\{ \ln x, \ln \frac{1}{x}\}$ on $(0,\infty)$

\item $\{ e^x , e^{x+7} \}$ on $(-\infty,\infty)$

\item $\{ \frac{x+1}{x^2-1}, \frac{x}{x^2-1} \}$ on $(-1,1)$
\end{enumerate}


{\bf III.}
For a set of three functions, $\{f_1, f_2, f_3 \}$, each twice differentiable, the Wronskian is defined to be

\[
\det \mat{f_1 & f_2 & f_3 \\ f'_1 & f'_2 & f'_3 \\ f''_1 & f''_2 & f''_3 }
\]
A set of three functions defined on an interval $I$ is linearly 
dependent if there exist real numbers, $C_1$, $C_2$ and $C_3$, not all zero such that
\[
C_1 f_1(x) + C_2 f_2(x) + C_3 f_3(x) = 0 \hspace{1in} (*)
\]
for all $x \in I$. Otherwise a set is linearly independent. 
It is known that if a set of three twice differentiable functions is linearly dependent
then the Wronskian is zero on $I$. ({\bf Extra Credit: Prove this!})

Find the Wronskian of the four sets below. If it is not always zero
conclude the set is linearly independent. If it is always zero
find values for $C_1$, $C_2$ and $C_3$, not all zero, that 
satisfy $(*)$. The interval for each is the whole real line.

\begin{enumerate}
\item $\{ \sin^2 x, \cos^2 x , \cos 2x \}$

\item $\{ \sin x , \sin 2x , \sin 3x \}$

\item $\{ x^2 + 3x, x + 2, x^2 + x - 4 \}$

\item $\{ x^2 , x^2 + x, x^2 + x + 1 \}$
\end{enumerate}


{\bf IV.} We consider differential equations of the form $y'' + p(t)y' + q(t)y = 0$.
Assume $p(t)$ and $q(t)$ are continuous on the whole real line.

1. Explain why both $y_1=t^3$ and $y_2 = t^4$ cannot be solutions. 

2. Explain why both $y_1=\sin t$ and $y_2=t^2+t$ cannot be solutions.

{\bf V.} 
1. Find a differential equation of the given form $y'' + p(x)y' + q(x)y = 0$ that has $\{x+1, x+2\}$ 
as a fundamental solution set. On what interval is it valid? 

2.  Find a differential equation of the given form  $y'' + p(x)y' + q(x)y = 0$ that has 
$\{\sin x, e^x \}$ as a fundamental solution set. On what interval is it valid? 

\end{document}
