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\noindent
\parbox{2.25in}{\bf Math 4010-101}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Apr. $6^\mathrm{th}$, 2022}

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

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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. More than Just Factoring:] Find the general solution of the following homogeneous linear differential equations:
\begin{enumerate}[(a)]
\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}
{\it Note:} I suggest handing off the algebra of multiplying out polynomials like $S(S-1)(S-2)$ etc. and factoring to software like Maple or a website like Wolfram Alpha.


\item[\#2. Making My Own Way Again:] 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)]
\item $ x^{-3}\ln(x)+x^5 $
\item $ x^{-1}\cos(3\ln(x)) $
\end{enumerate}

{\it Suggestion:} Find the factored polynomial in $S=t\frac{d}{dt}$ that annihilates these functions. Then subtract out $S(S-1)(S-2)$ (i.e., $x^3\frac{d^3}{dx^3}$) followed by subtracting out a multiple of $S(S-1)$ (i.e., $x^2\frac{d^2}{dx^2}$) etc. to find the Cauchy-Euler equation in the form $x^ny^{(n)}+a_{n-1}x^{n-1}y^{(n-1)}+\cdots+a_1xy'+a_0y=0$.

\item[\#3. 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 $ (sort of ``multiply it out'' but be careful about $x^2D \not= Dx^2$): $??y''+??y'+??y=0$. 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}

\item[\#4. Non-Homogeneous Problems:] You might want to use software to help differentiate, determine undetermined coefficients (i.e., solve equations), and factor characteristic polynomials.

\begin{enumerate}[(a)]
\item Consider $ y'''-3y'-2y=te^{-t}\sin(3t)-4e^{-t}+5t^2-2 $. Write down the most efficient ``guess'' for a particular solution as suggested by the method of undetermined coefficients. 

Don't worry about determining the coefficients and actually solving. Just write down the ``form'' for the particular solution.

\item Solve $ t^2y''+ty'+4y = \sin(\ln(t^2))+3\sqrt{t} $ using the method of undetermined coefficients.

\item Solve $ y''-4y'+3y=t^3+3 $ using\\ \force \qquad (1) the method of undetermined coefficients \\ \force \qquad (2) variation of parameters.

\end{enumerate}

\end{description}

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