### MAT 132 Spring 2006 Review - Chapter 7, Appendix I, Notes on 2nd order D.E.s

*References are to Stewart, Single Variable Calculus - SBU Edition*
7.2 Direction fields. Understand how the
direction field for the first-order differential equation
`y' = f(x,y)` allows one to "see" the family of
solutions [Figures 1-4 page 505]. Be able to sketch the
direction field of `y' = f(x,y)` [Example 1, Exercises
9,10, Exercises 11-14 -choose a 5 by 5 grid of points centered on the
initial value]. Given a direction field be able to sketch
solutions, and be able to sketch the solution with a given
initial value [Figures 7,8 page 506; Example 2; Exercises
1a, 3, 7, 18].

Euler's Method. Given a first-order differential
equation `y'=f(x,y)`, an interval `[0,T]`,
an initial value `y`_{0} and a number `N`,
be able to apply Euler's method with `N` steps to find
approximate values `y(T/N), y(2T/N), ... , y(T)`
for the solution satisfying `y(0) = y`_{0}
[Example 3 page 509, Example 4 page 510, Exercises 21-24].

7.3 Separable equations. Understand what a separable equation
is and be able to solve it by integration, applying the
"separate, integrate, solve" method [Examples 1a, 2 page 514].
Understand how the "`+C`" from
one of your integrals turns into the undetermined constant
in the general solution, and how an initial value determines
what that constant must be [Example 1b, Exercises 9-14].

Mixing problems. Be able to apply the
"rate in - rate out" method to convert a mixing problem to
a separable differential equation, and then be able to solve the
equation [Example 6, problems 35. 37].

7.4 Exponential growth and decay. `y' = ky` is an important
special case of a separable equation. The general solution
is `y=Ce`^{kt}, called "exponential growth" if
`k` is positive, "exponential decay" if `k` is
negative. Understand that `C = y(0)` [Exercises 1,2].
Be able to calculate `k` from `C` and one other
data point [Exercise 3].
Be able to calculate `C` and `k` from two data points
[Exercise 4]. Special case: the half-life. Be able to calculate
`k` from the half-life, and vice-versa [Example 3, Exercises
8, 9, 10].

Compound interest. Understand that $1 invested for
one year at annual interest t (in dollars) gives $1+t if compounded annually,
$(1 + t/2)^{2} if compounded twice a year
$(1 + t/n)^{n} if compounded n times a year,
and $e^{t} if compounded continuously [Example 5 page 531,
Exercises 18a, 19a].

Complex Numbers - Appendix I. Understand the identification
of the complex number `x + iy` with the point `(x,y)`
in the plane. Be able to add and multiply complex numbers.
[Exercises 1, 2, 3, 4].
Be able to write the complex number `x + iy` in "polar"
form `r e`^{i theta}. [Box, p.A72, Example 4 p.A73,
Exercises 25, 26]. Understand the geometric explanation of
multiplication (multiply the `r`'s and add the `theta`'s)
[Equato 1, p.A73 and the sentence following].

Second Order Differential Equations (we only study an important
special case: linear homogeneous equations with constant coefficients).
Review the Notes. Be able to
substitute `y = e`^{lambda x} for the unknown
function `y` and to solve for `lambda`. This
method gives two solutions (remember how to get them in the special case
where the `lambda` equation has a double root),
say `y`_{1}
and `y`_{2}. Be able to write the general solution
as `c`_{1}y_{1} + c_{2}y_{2}
[Exercises 1, 2 in Notes]
and to use initial conditions to calculate `c`_{1}
and `c`_{2} [Exercises 6, 9 in Notes]. The
fundamental physical example is a horizontal spring fixed at
one end and with a mass of 1 gram at the other. Then, taking `y = 0`
as the rest position, the position
of the mass is given by the solution of `y'' + py' + ky = 0`
where `k` is the spring constant and `p` is the
damping constant [Exercises 11-15]