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{\Large TEST 1

Math 2280 - 7}
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Instructions:
\begin{itemize}
\item  Due: Beginning of Class, Monday July 10.
\item  Show all your work.  
\item  You may use your book, notes and calculator but you cannot you anyone else or a computer.
\item  You have several days to do this, so what you turn in should be neat.
You will be marked down if I have trouble reading or following your work.
\end{itemize}


\begin{enumerate}



\item  (12 points) For each of the initial value problems below, 
determine 
\begin{enumerate}
\item[(i)]  If there is a solution to the problem, and if there is the interval the solution will be valid.
\item[(ii)]  If this is the unique solution.
\item[(iii)]  Tell me your answers in the previous parts is correct.
\item[(iv)]  If you are unable to determine the above, then explain why not.
\end{enumerate}
\begin{enumerate}
\item
\[
\frac{dy}{dx}=\sqrt{2x-y} \quad \quad y(1)=2
\]
\item 
\[
\frac{dy}{dx}=\frac{x^4y+1}{2x^2-77x+488} \quad \quad y(10)=6
\]
\item
\[
y' = (2x-y)^{2/3} \quad y(1)=-2
\]
\item 
\[
y^{(4)}+\sqrt{x+1}y^{(3)}-\frac{x}{x^2-4}y''+6e^xy=0 \quad \quad y(0)=4
\]
\end{enumerate}


\item  (10 points) Find the general solution to the differential equation
\[
\frac{6}{x^3} \frac{dy}{dx} = y
\]

\item  (10 points) Solve the initial value problem
\[
xy' -6y =x^4 \quad y(2) = 24
\]

\item  (10 points) Find the general solution to the differential equation below:
\[
y''-y'-2y=e^{3x}
\]
Hint: First solve the homogeneous equation, then to find a particular solution, 
try $Ae^{3x}$ and see if you can find $A$.



\item  (10 points) Prove that the functions below are linearly independent
\[
f(x)=x \quad g(x)=xe^x \quad h(x)=x^2e^x
\]


\item  (15 points) A hovering helicopter's engines suddenly dies 2000 feet above the ground.
The helicopter plummets to the ground.
\\
Assume that the air resistance is proportional to the velocity of the helicopter with $\rho=.18$
(so that up to sign we have $F_R=m\rho v$).
Find the following:
\begin{enumerate}
\item  The terminal velocity of the helicopter,
\item  The amount of time that it takes the helicopter to hit the ground,
\item  The speed that the helicopter is going when it hits the ground.
\end{enumerate}
You MUST set up and solve the differential equations to get full credit 
(don't use the book's general solution).

\item (12 points) Suppose that a certain species of fish in the Deer Creek Reservoir satisfy the differential equation
\[
\frac{dP}{dt} = -\frac{1}{2}P^3+\frac{21}{2}P^2-72P+160
\]
where the units are in hundreds of fish per year.
\begin{enumerate}
\item  Find all equilibrium solutions and label them as stable, unstable or semi-stable,
\item  draw the phase diagram,
\item  draw a graph of possible solution curves for this differential equation.
\item  If there are initially 100 fish in the reservoir, what happens over time?
\item  If 20,000 fish were suddenly planted in the reservoir (a one time occurrence) what would happen over time?
\end{enumerate}
Hint: get your units right!  Do you remember how to factor polynomials?


\item (10 points)  Consider the initial value problem
\[
\frac{dy}{dx} = e^{2y} \cos (xy) \quad y(1)=2
\]
Using Runge-Kutta and a step size of $0.5$, find an approximation for $y(0)$.

\item  (12 points)  Suppose we have the differential equation
\[
y''+p(x)y'+q(x)y=0
\]
and we have two solutions $y_1$ and $y_2$.
Make the following definitions:
\[
y_3:=y_1+y_2 \quad y_4:=y_1-y_2
\]
Prove that if $y_1$ and $y_2$ are linearly independent solutions then 
$y_3$ and $y_4$ are linearly independent solutions.






\end{enumerate}

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