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Math 305 \hfill Homework Set 5  \hfill Spring 2017

\begin{center}Due Monday February 20\end{center}

{\bf I.} For each differantial equation below, find the general 
solution, the particular solution for the initial values given, and 
graph it. (You can use a computer or calculator, but label the 
intescepts and roughly the value of any extrema.)

\begin{enumerate}
\item $y'' - y' -2y = 0$, $y(0)=1$, $y'(0) = 2$.
\item $4y'' - 4y' + y = 0$, $y(1) = 2$, $y'(1) = -1$.

\item $y'' -4 y' + 13 y = 0$, $y(0)=4$, $y'(0) = -1$.
\item $y'' + 9y = 0$, $y(0)=2$, $y'(0) = 1$.
\end{enumerate}

{\bf II.} These involve equations with nonconstant coefficients.

1. Consider $t^2 y'' + 4t y' - 10 y = 0$. Suppose that $y(t) = t^r$ is a solution. 
What are the allowed values of $r$? What do you think the general solution is? Plug it in and check. 
Find the partictular solution if $y(1) = 2$ and $y'(1) = 3$. On what interval do you think it will be valid?


\vspace{.25in}

2. Consider $y'' - \frac{1}{x} y' + 4 x^2y = 0$. Assume $x>0$. (a) Show that $y_1(x) = \sin(x^2)$ 
is a solution. (b) Let $y_2(x) = v(x)\sin(x^2)$. Use the 
reduction of order method to find another solution that is not a multilple of $y_1$.



\vspace{.25in}


3. Consider $t^2 y'' - t y' + y = 0$. Suppose $t^r$ is a solution. Find $r$. Use the Reduction of Order Method, as in \#2, to find a 
second solution. What is the general solution?


\vspace{.25in}

{\bf III.} [Extra Credit]. Consider $y' = t(5-y)$, $y(0)=0$. Find the solution. Use a 
spreadsheet program to compare the solution with the 
Euler Method and Improved Euler Method approximations for stepsize $h=0.1$ for 
$t_n = nh$, from $0.0$ to $5.0$.
 
See the spreadsheet link next to this hwk link for an example where the equation was $y'=t-2y$. 

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