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\parbox{2.25in}{\bf Math 4010-101}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Mar. $30^\mathrm{th}$, 2022}

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%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force


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\begin{description}
\item[\#1. First Order Problems:] %(Rabenstein 1.4 \#6 \& \#8 page 28 both 2nd and 3rd editions) 
Solve the following initial value problems.
\begin{enumerate}[(a)]
\item $ y' +(\cos(x))y=\cos(x) $, $ y(\pi)=0 $
\item $ (x^2+1)y'-2xy=x^2+1 $, $ y(1)=\pi $
\end{enumerate}


\item[\#2. Lowering Orders:] Convert the following systems of differential equations to an equivalent first order system (using $x_1,x_2,\dots $ as dependent variable names). 
State whether the system is linear or not. If the system is linear, write the equivalent first order system in vector-matrix form: ${\bf x}'(t) = A(t){\bf x}+{\bf g}(t).$
\begin{enumerate}[(a)]
\item $ (y'')^3+\sin(y'z')=e^{y+z''} $ and $ z'''+e^{2t}y'+\sin(t)z'=5 $
\item $ y'''+t^5y''+e^{-t^2}y'+\sin(t)y=t^3-7 $
\end{enumerate}

\item[\#3. Just Factoring:] Find the general solution of the following homogeneous linear differential equations:
\begin{enumerate}[(a)]
\item $ y^{(5)}-4y'''=0 $
\item $ y^{(6)}+6y^{(5)}+16y^{(4)}+32y'''+48y''+32y'=0 $
\item $ y^{(4)}+4y'''+24y''+40y'+100y=0 $

%
% These will probably be part of Homework #7
%
%\item $ x^3y'''+x^2y''-6xy'+6y=0 $
%\item $ x^4y^{(4)}+10x^3y'''+32x^2y''+54xy'+36y=0 $
%\item $ x^3y'''+8x^2y''+23xy'+13y=0 $
\end{enumerate}

\item[\#4. Making My Own Way:] %(Rabenstein 5.5 \#29 b,d,f page 207 in 2nd edition and page 236 in 3rd edition) 
Find a linear homogeneous differential equation with real constant coefficients, whose order is as low as possible, that has the given function as a solution:
\begin{enumerate}[(a)]
\item $ x-e^{3x} $
\item $ e^x\sin(2x) $
\item $ \cos(2x)+3e^{-x} $
\end{enumerate}

%
% These will probably be part of Homework #7
%
%Find a linear homogeneous differential equation of Cauchy-Euler type (with real coefficients), whose order is as low as possible, that has the given function as a solution.
%\begin{enumerate}[(a)]
%$ x^{-3}\ln(x)+x^5 $
%$ x^{-1}\cos(3\ln(x)) $
%\end{enumerate}

%
% These will probably be part of Homework #7
%
%\item[\#5. Fun with Differential Operators:] For convenience, let $a,b,c$ be constants and $D=\frac{d}{dx}$.
%\begin{enumerate}[(a)]
%\item The differential operators $ x^2D $ and $ Dx^2 $ are not equal. Find an (operator) equation relating them.
%\item The equation $ (x^2D-a)[y]=0 $ is the same as $ x^2y'-ay=0 $. Find the general solution of this equation. Separation of variables should work.
%\item Find the second order linear equation equivalent to $ (x^2D-a)(x^2D-b)[y]=0 $. Find the general solution of such an equation. Consider 3 cases: (1) Distinct real roots, (2) A repeated real root, and (3) A pair of complex conjugate roots.
%\item Find the third order linear equation equivalent to $ (x^2D-a)^3[y]=0 $. Also, find the general solution of this equation.
%\end{enumerate}

\end{description}

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