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\begin{document}
\begin{center} \large\sf Week 1 Examples\end{center}
{\bf Three Examples}. Solve differential equations without a book. Three basic examples used throughout a course in differential equations, which require only a calculus background.
\begin{quote}
{\bf Growth-Decay}: $\dd\frac{dA(t)}{dt} = k \, A(t)$, $A(0)=A_0$.\newline The unique solution is $A(t) = A_0\, e^{kt}$. {\small\sf Radioactive decay. Jeweler's bench light experiment. Malthusian population dynamics.
RC and LR circuits. Drug elimination. First-order chemical reactions, law of mass-action. Compound continuous bank interest.}

{\bf Newton Cooling}: $\dd\frac{du(t)}{dt} = -h(u(t)-u_1)$, $u(0)=u_0$.\newline The solution is $u(t) = u_1+(u_0-u_1)\, e^{-ht}$. {\small\sf Hot chocolate at initial temperature $u_0$ with room thermometer reading $u_1$. Symbol $u(t)$ = time-varying hot chocolate dial thermometer temperature.}

{\bf Verhulst Dynamics}: $\dd\frac{dP(t)}{dt} = (a-b\, P(t))\, P(t)$, $P(0)=P_0$.\newline The solution is $P(t) = \dd\frac{aP_0}{bP_0+(a-bP_0)e^{-at}}$. {\small\sf Fish population $P(t)$ in {\bf Cecret Lake} at {\bf Alta}. Carrying capacity. Stocking and re-stocking. Harvesting.}
\end{quote}
{\bf Example 1}: Exercise 1.2-2: Solve $dy/dx=(x-2)^2$, $y(2)=1$.\newline
{\small\sf Method of quadrature. Answer Check details. Non-reversible steps, false proof for $0=1$.}\newline
\href{http://www.math.utah.edu/~gustafso/s2015/2250/FTC-Method-of-Quadrature.pdf}{http://www.math.utah.edu/~gustafso/s2015/2250/FTC-Method-of-Quadrature.pdf}

{\bf Example 2}: Exercise 1.2-4:  $dy/dx = 1/x^2, y(1)=5$.\newline
{\small\sf Power rule in Newton calculus. Answer check shortcuts.}

{\bf Example 3}: Exercise 1.2-10:  $dy/dx = x \, e^{-x}, y(0)=1$.\newline
{\small\sf Integral tables and integration by parts. Jennifer Lahti's solution:\newline
\href{http://www.math.utah.edu/~gustafso/s2015/2250/2250Week1exercises-JenniferLahti-1.2-5+8+10.pdf}{http://www.math.utah.edu/\~{}gustafso/s2015/2250/2250Week1exercises-JenniferLahti-1.2-5+8+10.pdf}

{\bf Example 4}: Exercise 1.3-8: $dy/dx = x^2 - y$\newline
Thread edge-to-edge solutions through the direction field at each blue dot. JPEG image source:
\newline
\href{http://www.math.utah.edu/~gustafso/s2015/2250/exercise1.3-8-EdwardsPenney.jpg}{http://www.math.utah.edu/\~{}gustafso/s2015/2250/exercise1.3-8-EdwardsPenney.jpg}

{\bf Example 5}: Exercise 1.3-14: $dy/dx = y^{1/3}, y(0)=0$\newline
Explain application of the Peano and Picard theorems. \newline
{\small\sf Computer numerical methods fail on this example. Why?}

{\bf Example 6}: Exercise 1.4-6:  Solve $y'=3 \sqrt{xy}$\newline
{\small\sf  Three answers. Book reports only one answer. }

{\bf Example 7}: Exercise 1.4-10: Solve $(1+x^2)y' = (1+y)^2$\newline
{\small\sf Two answers. Book reports only one answer.}

{\bf Example 8}: Exercise 1.4-18:  Solve $x^2y' = 1 - x^2 + y^2 - x^2y^2$ [See Example 11 {\em infra}]

{\bf Example 9}: Exercise 1.4-22: Solve $y'=4x^3y-y, y(1)=-3$


{\bf Example 10}: Show that $y'=x+y$ is not separable.\newline
{\small\sf  TEST I. $f_x/f$ depends on $y$ implies $y'=f(x,y)$ not separable.}

{\bf Example 11}: Find a factorization $f(x,y)=F(x)G(y)$ given
\\
       (1) $f(x,y)=2xy+4y+3x+6$ \\
       (2) $f(x,y)=(1-x^2+y^2-x^2y^2)/x^2$  \\
{\small\sf Answers: (1) $F=x+2, G=2y+3$; (2) $F=(1-x^2)/x^2, G=1+y^2$. Main idea: Choose $y=0$ in $F(x)=f(x,y)/G(y)$ to find $F(x)=(3x+6)/G(0)$ in equation (1). How to find $G$? {\bf Warning}: Divide by zero is not allowed. Choose $y=0$, $y=1$, etc, until no divide by zero error.}

{\bf Example 12}: Midterm 1 examples:
    $y'=x+y, y'=x+y^2, y'=x^2+y^2$
\end{document}
