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\noindent
\parbox{2in}{\bf Math 3130}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{2in}{\bf \hfill Due: June $25^\mathrm{th}$\!,\  2019}

\noindent \force \hfill {\bf Be sure to show your work!} \hfill \force

\vspace{0.15in}

You may use technology to complete this assignment. Although much of it {\it could} be done by hand.

\vspace{0.15in}

\noindent {\bf\large 1.} Solve the initial value problems.\\ \force \qquad {}[Do this ``by hand'' using the method of undetermined coefficients. Technology can help confirm answers.]
\begin{enumerate}[(a)] 
\item $y''+9y=6\sin(t)$ with $y(0)=1$ and $y'(0)=-1$. 
\item $y''+9y=6\sin(3t)$ with $y(0)=1$ and $y'(0)=-1$. 
\item Graph both solutions. Even though the IVPs are very similar, the solutions look very different. Why?
\end{enumerate}

\vspace{0.15in}

\noindent {\bf\large 2.} Find the general solution of $y''+3y'+y=5\cos(3t)$. Write your solution in real form (no complex stuff). In addition, the forced response (the particular solution coming from the forcing function) is a combination of $\sin(3t)$ and $\cos(3t)$. Rewrite that part of the solution in the form: $A\cos(3t+\phi)$. Your $A$ should be exact. Round your $\phi$ to 3 decimal places.

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\noindent {\bf\large 3.} Consider the system $x'=x(2-x-y)$ and $y'=y(y-x^2)$. 
\begin{enumerate}[(a)]
\item Find the equilibria for this system. Use the method of almost linear systems (i.e., linearization) to analyze these equilibria. Notice that $(x,y)=(0,0)$ is one of the equilibria. The linearization technique doesn't help here. Why?
\item Plot a phase portrait of this system (pick a domain big enough to display all equilibria).
\item Sketch nullclines (with arrows indicating direction as done in Section 5.2).
\end{enumerate}


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