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\begin{document}
\title{Some \TeX/\LaTeX~Snippets from Homework 1, CPSC 303, Spring 2024}

\author{Joel Friedman}
\address{Department of Computer Science, 
        University of British Columbia, Vancouver, BC\ \ V6T 1Z4, CANADA.}
\curraddr{}
\email{{\tt jf@cs.ubc.ca}}
\thanks{Research supported in part by an NSERC grant.}
\urladdr{http://www.cs.ubc.ca/~jf}

\maketitle                 % Produces the title.

\noindent
{\bf Copyright:\ \ }
Copyright Joel Friedman 2024.  
Not to be copied, used, or revised without explicit written
permission from the copyright owner.

\ \bigskip

\begin{enumerate}
\item
(2 points)
\begin{enumerate}
\item
Show that the linear system
$$
\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 1 & 4 \end{bmatrix} 
\begin{bmatrix} c_0 \\ c_1 \\ c_2 \end{bmatrix}
= \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}
$$
has the unique solution
$$
\begin{bmatrix} c_0 \\ c_1 \\ c_2 \end{bmatrix}
=
\begin{bmatrix} -3/2 \\ 4/2 \\ -1/2 \end{bmatrix}
$$
\item
Say that $f\from\reals\to\reals$ has $f'''(x)$ existing for all $x$.
Say that $x_0,h\in\reals$, and that
$|f'''(\xi)|\le M_3$ for all $\xi$ bewteen $x_0$ and $x_0+2h$.
Use the fact that 
\begin{align*}
f(x_0) & = f(x_0) \\
f(x_0+h) & = f(x_0) + h f'(x_0) + \frac{h^2}{2} f''(x_0) + O(h^3)M_3 \\
f(x_0+2h) & = f(x_0) + 2h f'(x_0) + \frac{(2h)^2}{2} f''(x_0) + O(h^3)M_3 
\end{align*}
to find a value of $c_0,c_1,c_2$ such that
$$
c_0 f(x_0) + c_1 f(x_0+h) + c_2 f(x_0+2h) = h f'(x_0)+ O(h^3)M_3.
$$
\item
To which formula on page 411 (Section~14.1) of [A\&G]
are parts~(a) and~(b) related?  Explain.
\item
What is the significance of the solution of
$$
\begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 4 & 9
\\ 0 & 1 & 8 & 27 \end{bmatrix} 
\begin{bmatrix} c_0 \\ c_1 \\ c_2 \\ c_3 \end{bmatrix}
= \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}
$$
to approximating $f'(x_0)$?
[You don't have to solve this system, just state what you can
do with the solution $c_0,c_1,c_2,c_3$.]
\end{enumerate}

\bigskip\bigskip 

\item
(2 points)
Consider an ODE
$y'=f(t,y)$, where as in [A\&G], $y=y(t)$, 
and $y'$ refers to $dy/dt$.
Say that $f(t,y)$ is of the special form $f(t,y)=h(t)g(y)$, where
$g$ is differentiable function and $h$ is continuous.
Then the ODE
$$
y' = dy/dt = h(t) g(y)
$$
is called a {\em separable differential equation}, and it can be solved
by writing 
$$
\frac{dy}{g(y)} = h(t)\,dt,
$$
and taking indefinite integrals of both sides.
See
\href{https://personal.math.ubc.ca/~CLP/CLP2/clp_2_ic/sec_sep_de.html}{Section 2.4 (Separable ODE's) of UBC's Calculus 2 Textbook}
for details, including Example~2.4.2 there,
where they solve the equation $y'=y^2$
(in this textbook, $y'$ refers to $dy/dx$, as is common in math books).
\begin{enumerate}
\item
Solve the ODE $y'=y^3$ (here $y=y(t)$ and $y'$ refers to $dy/dt$)
in the same manner as $y'=y^2$ is solved in 
in general form.
\item Solve $y'=y^3$ for the initial condition $y(1)=1$.
\item Solve $y'=y^4$ for the initial condition $y(1)=1$.
\item Let $y(t)$ be as in part (b); for $t\ge 1$, when does $y(t)$ fail
to exist, i.e., for which $T>1$ does $y(t)\to\infty$ as $t\to T$?
\item Same question for part~(c).
\end{enumerate}

\bigskip\bigskip

\item
(2 points)
Let $y(t)=(3-2t)^{-1/2}$, $z(t)=(4-3t)^{-1/3}$.
\begin{enumerate}
\item
Examine a plot of $y(t)$ and $z(t)$ for $1\le t < 4/3$.
Is one of these functions larger than the other in the entire interval
(as far as the plot shows)?
(Here a simple answer will do.
You might type 
\begin{verbatim}plot (3-2t)^(-1/2) and (4-3t)^(-1/3)\end{verbatim}
into
Google, or someting like that.)
\item
Show that $y'=y^3$ and $z'=z^4$ and that
$y(1)=z(1)=1$.
\item
Show that $y'(1)=z'(1)=1$.
\item
Show that $y''(1)=3$ and $z''(1)=4$.
[Hint: differentiate both sides of $y'=y^3$; similarly for $z$.]
\item
Show that for $h>0$ and $h$ sufficiently small we have
$y(1+h)<z(1+h)$.
[Hint: let $u(t)=z(t)-y(t)$; what are the values of $u(1),u'(1),u''(1)$ ?]
\end{enumerate}


\end{enumerate}

\end{document}

