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Math 2214. Test 1

1. Solve the initial value problem

\begin{displaymath}ty'+2y=t^3, y(2)=1.\end{displaymath}

On which interval does the solution exist? How does it behave at the ends of this interval?
2. Solve the initial value problem

\begin{displaymath}y'=y^2+y, y(0)=1,\end{displaymath}

What is the maximal interval on which the solution exists and how does the solution behave at the ends of this interval?
3. Sea water has a salt content of about 35 grams per liter, while river water has about 0.12 grams per liter. An aquarium that used to have saltwater fish is to be used for river fish. The aquarium initially contains 200 liters of sea water, and it is gradually replaced with river water at a rate of 2 liters per minute, while the well mixed solution leaves the aquarium at the same rate. It is safe to stop the refilling and put the fish in when the salt concentration is down to 0.15 grams per liter. How long does it take to reach the desired level?
4. Use the Euler method with $h=0.1$ to find an approximation for $y(1.2)$, where $y(t)$ satisfies the equation

\begin{displaymath}y'=t+y^2,\quad y(1)=2.\end{displaymath}

Answers:

1.

\begin{displaymath}y={t^3\over 5}-{12\over 5t^2}.\end{displaymath}

The solution exists for $0<t<\infty$, $y\to -\infty$ for $t\to 0$, $y\to\infty$ for $t\to\infty$.

2.

\begin{displaymath}y={e^t\over 2-e^t}.\end{displaymath}

The solution exists for $-\infty<t<\ln 2$, $y\to 0$ for $t\to -\infty$, $y\to\infty$ for $t\to\ln 2$.

3.

\begin{displaymath}Q'(t)=0.24-{Q(t)\over 100},\quad Q(0)=7000.\end{displaymath}

The solution is

\begin{displaymath}Q(t)=24+6976e^{-t/100}.\end{displaymath}

The desired state $Q=30$ is reached when $t=100\ln(3488/3)=705.85$.

4. $y_1=2.5$, $y_2=3.235$.


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Next: About this document ... Up: 2214samp1 Previous: 2214samp1
Michael Renardy 2004-09-26