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1.1:  Examples of differentiable equation:

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1.)  $F = ma = m {dv \over dt} = mg - \gamma v$
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2.)  Mouse population increases at a rate proportional to the current population:

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More general model :  ${dp \over dt} = rp - k$ \hfil \break
where $p(t)$ = mouse population at time $t$,\hfil \break
 $r$ = growth rate or rate constant, \hfil \break
 $k=$ predation rate = \# 
mice killed per unit time.

%%3.)  Continuous compounding ${dS \over dt} = rS + k$ \hfil \break
%%where $S(t) = amount of money at time $t$, $ $r$ = interest rate, \hfil \break
%% $k=$ constant deposit rate 

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direction field = slope field = graph of ${dv \over dt}$ in $t, v$-plane.

%%See direction field created using \hfil
%% \break
 www.math.rutgers.edu/$\sim$ sontag/JODE/JOdeApplet.html

*** can use slope field to determine behavior of $v$ including as $t 
\rightarrow \infty$.

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Equilibrium Solution = constant solution

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1.2:  Solve ${dy \over dt} = ay + b$ by separating variables:

${dy \over ay + b} = dt$

$\int {dy \over ay + b} = \int dt$
~~~~~implies~~~~~
${ln|ay + b| \over a} = t + C$


${ln|ay + b|} = at + C$
~~~~~implies~~~~~
$e^{ln|ay + b|} = e^{at + C}$


$|ay + b| = e^C e^{at}$
~~~~~implies~~~~~
$ay + b = \pm(e^C e^{at})$

$ay  = C e^{at} - b$
~~~~~implies~~~~~
$y  = C e^{at} - {b \over a}$


Initial Value Problem:  $y(t_0) = y_0$
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1.3:
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ODE (ordinary differential equation):  single independent variable
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~~~Ex:  ${dy \over dt} = ay + b$
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   vs 

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PDE (partial differential equation):  several independent variables
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~~~Ex:  ${\partial xy \over \partial x} = {\partial xy \over \partial y}$

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order of differential eq'n:  order of highest derivative
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example of order $n$:  $y^{(n)} = f(t, y, ..., y^{(n-1)})$

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Linear vs Non-linear
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linear:  $a_0(t) y^{(n)} + ... + a_n(t)y = g(t)$

Determine if linear or non-linear:
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Ex:  $t y'' - t^3y' - 3y = sin(t)$
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Ex:  $ 2y'' - 3y' - 3y^2 = 0$

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********Existence of a solution**************
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********Uniqueness of solution***************

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CH 2:  Solve ${dy \over dt} = f(t, y)$

2.2:  Separation of variables: $N(y)dy = P(t)dt$

2.1:  First order linear eqn:  ${dy \over dt} + p(t)y = g(t)$


Ex 1:  $t^2y' + 2ty = t sin(t)$

Ex 2:  $y' = ay + b$

Ex 3:  $y' + 3t^2y = t^2$, $y(0) = 0$

Note:  could use section 2.2 method, separation of variables to solve ex 2 and 3.

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Ex 1:  $t^2y' + 2ty = sin(t)$ \hfil \break
(note, cannot use separation of variables).

$t^2y' + 2ty = sin(t)$

$ (t^2y)' = sin(t)$

$ \int (t^2y)' dt = \int sin(t) dt$

$ (t^2y) = -cos(t) + C$
implies
$ y = -t^{-2}cos(t) + Ct^{-2}$
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Gen ex:  Solve $y' + p(x)y = g(x)$

Let $F(x)$ be an anti-derivative of $p(x)$

$e^{F(x)} y' + [p(x) e^{ F(x)} ] y = g(x)e^{F(x)}$

$e^{F(x)} y' + [F'(x) e^{ F(x)} ] y = g(x)e^{F(x)}$

$[e^{F(x)} y]' = g(x)e^{F(x)}$


$e^{F(x)} y = \int g(x)e^{F(x)} dx$

$ y = e^{-F(x)} \int g(x)e^{F(x)} dx$







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