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\noindent
\parbox{2.25in}{\bf Math 4010-101}
\hfill {\Large \bf Homework \#8} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Apr. $13^\mathrm{th}$, 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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\noindent {\bf Assumption:} For Cauchy-Euler equations, we assume that the independent variable is positive (i.e., $x>0$ or $t>0$).

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\begin{description}
\item[\#1. Solving the World's Problems:] You \sout{might} will want to use software to help with some parts.

\begin{enumerate}[(a)]
\item Solve $y''-4y'+3y=t^3+3$ using the method of undetermined coefficients, variation of parameters, and by converting to an equivalent linear system.

{\it Note:} We already solved this equation using undetermined coefficients and variation of parameters in Homework \#7. But enough people didn't get it right, so let's try again.

\item Solve $y'''-5y''+8y'-4y=e^t$ using the method of undetermined coefficients, variation of parameters, and by converting to an equivalent linear system.
\item Solve $t^2y''+2ty'-6y = e^{2t}$ then find the solution whose initial values are $y(1)=2$ and $y'(1)=3$.
\item Solve $y'''-y''+9y'-9y = g(t)$ where $g(t)$ is a continuous function.
\end{enumerate}

\item[\#2. Systemic Problems:] Again, I encourage you to use software to aid in calculations. 

\begin{enumerate}[(a)]
\item Find the general solution of the system: $\begin{array}{ccccc} x_1' & = & -x_1 & & \\ x_2' & = & -x_1 & & +3x_3 \\ x_3' & = & 3x_1 & -3x_2 & \end{array}$
 
     Then solve the corresponding initial value problem where $x_1(0)=0, x_2(0)=1, x_3(0)=1$.
\item Find the general solution of the system: $\begin{array}{cccccc} x_1' & = & x_1 & +x_2 & +x_3 & +1 \\ x_2' & = &  & -x_2 & & \\ x_3' & = & -2x_1 & -x_2 & -2x_3 & +2e^{-t} \end{array}$

     Then solve the corresponding initial value problem where $x_1(0)=3, x_2(0)=2, x_3(0)=1$.
\item Find the general solution of the system: $\begin{array}{cccccc} x_1' & = & x_1 & -x_2 & +3x_3 & -2t \\ x_2' & = & -x_1 & +2x_2 & +2x_3 & \\ x_3' & = & -x_1 & & +4x_3 & +2t \end{array}$ 

     Then solve the corresponding initial value problem where $x_1(0)=0, x_2(0)=1, x_3(0)=0$.
\end{enumerate}

\end{description}

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