Math 104

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Math 104
July 26
Differential Equations
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Arc length
The length of a curve in the plane is generally
difficult to compute. To do it, you must add up
the little pieces of arc, ds. A good
approximation to ds is given by the Pythagorean
theorem We can use this to find the length
of any graph provided we can do the integral
that results!
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Example
Find the arclength of the parabola y x2 for x
between -1 and 1. Since dy / dx 2x, the
element of arclength is
so the total length is
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Conclusion
So far, we have that the length is
To do this integral, we will need a trig
substitution. But, appealing to Maple, we get that
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Can we do the integral?
The arc length integral from before was
This is a trig substitution integral of the
second kind
With the identity tan2 1 sec2 in mind, let
What about dt ? Since we
have that
These substitutions transform the integral into
This is a tricky integral we need to do by parts!
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To integrate
Let
Then
But tan2 sec2 - 1 , so rewrite the last
integral and get
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Still going.
Its remarkable that were almost done. The
integral of secant is a known formula, and then
you can add the integral of sec3 to both sides
and get
So weve got this so far for
where
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We need a triangle!
So far,
with
From the triangle,
So
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Definite integral
So far, we have
Therefore
To get the answer Maple got before, wed have to
rationalize the numerator inside the logarithm.
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Surface Area
The area of a surface of revolution is calculated
in a manner similar to the volume. The following
illustration shows the paraboloid based on (for
x0..2) that we used before, together with one of
the circular bands that sweep out its surface
area.
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To calculate the surface area
To calculate the surface area, we first need to
determine the area of the bands. The one
centered at the point (x,0) has radius
and width equal to .
Since we will be integrating with respect to x
(there is a band for each x), we'll factor the dx
out of ds and write .
So the area of the band centered at (x,0) is
equal to Thus, the total surface area is
equal to the integral
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The surface area turns out to be
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p
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Differential Equations

• The most important application of integrals is to
  the solution of differential equations.
• From a mathematical point of view, a differential
  equation is an equation that describes a
  relationship among a function, its independent
  variable, and the derivative(s) of the function.

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For example
ORDER highest derivative first order, second
order...
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To solve a differential equation
means to find a function y(x) that makes it true.
solves
solves
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In Applications

• Differential equations arise when we can relate
  the rate of change of some quantity back to the
  quantity itself.

y
dy/dx
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Example (1)
The acceleration of gravity is constant (near the
surface of the earth). So, for falling objects
the rate of change of velocity is constant
Since velocity is the rate of change of position,
we could write a second order equation
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Example (2)
Here's a better one -- with air resistance, the
acceleration of a falling object is the
acceleration of gravity minus the acceleration
due to air resistance, which for some objects is
proportional to the square of the velocity. For
such an object we have the differential equation
rate of change of velocity is
gravity minus something proportional to
velocity squared
or
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Example (3)
In a different field Radioactive substances
decompose at a rate proportional to the amount
present. Suppose y(t) is the amount present at
time t.
rate of change of amount is
proportional to the amount (and decreasing)
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Other problems that yield the same equation
In the presence of abundant resources (food and
space), the organisms in a population will
reproduce as fast as they can --- this means that
the rate of increase of the population
will be proportional to the population itself
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..and another
The balance in an interest-paying bank account
increases at a rate (called the interest rate)
that is proportional to the current balance. So
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More realistic situations for the last couple of
problems
For populations An ecosystem may have a maximum
capacity to support a certain kind of organism
(we're worried about this very thing for people
on the planet!). In this case, the rate of
change of population is proportional both to the
number of organisms present and to the amount of
excess capacity in the environment (overcrowding
will cause the population growth to decrease).
If the carrying capacity of the environment is
the constant Pmax , then we get the equation
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and for the Interest Problem...
For annuities Some accounts pay interest but at
the same time the owner intends to withdraw money
at a constant rate (think of a retired person who
has saved and is now living on the savings).
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Question
If the interest rate is r , and the retiree wants
to withdraw W dollars per year, which is the
correct differential equation for the balance B
in the account at time t?
A)
D)
B)
E)
C)
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Another application
According to Newton's law of cooling , the
temperature of a hot or cold object will change
at a rate proportional to the difference between
the object's temperature and the ambient
temperature. If the ambient temperature is kept
constant at A, and the object's temperature is
u(t), what is the differential equation for u(t)
?
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Solving Differential Equations
Since the the process of solving of a
differential equation recovers a function from
knowing something about its derivative, it's not
too surprising that we have to use integrals to
solve differential equations. And since were
using integrals, we should also expect to see
some "arbitrary" constants in the solutions of
differential equations. In general, there will be
one constant in the solution of a first-order
equation, two in a second-order one, etc...
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In practice...
In practice, we can solve for the constants by
having some information about the value of the
unknown function (and/or the value of its
derivative(s)) at some point. From an
applications point of view, such initial
conditions are clearly needed, since you can't
determine the value of something just from
information about how it is changing. You also
need to know its value at some ("initial") time.
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Some examples will make this clear
Let's go back to the very first example, This
is an example of a separable first-order equation
(the only kind we'll worry about today). If you
view dy and dx as variables (so you can multiply
both sides by dx), you can get all the x's on one
side and all the y's on the other by algebraic
manipulation. Here, you can write
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Equation of differentials...
This is an actual "equation of differentials".
Then, simply integrate both sides
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(You only need one constant of integration).
This is called the "general solution" of the
differential equation. We can determine C if we
were given one point on the graph of the function
y(x). For instance, if you were given that
y(1)2 , then you could substitute 2 for y and 1
for x and get and so you would conclude that C
-2, so the solution of the initial-value
problem is ,
or (better)
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DiffEq Problem
If the function y f (x) satisfies the
initial-value problem then f (1)
A)
E)
B)
F)
C)
G)
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D)
H)
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DiffEq Greatest Hits
Hit 1 The water in the tank problem!
A tank contains 1000 liters of brine (salty
water) with 50 kg of dissolved salt. Pure water
enters the tank at the rate of 25 liters per
minute, The solution is kept thoroughly mixed and
drains at an equal rate. How many kg of salt
remain in the tank after 10 minutes?
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The setup.
The first step in most DiffEq problems is to
identify the unknown function. Since we want to
know the amount of salt at different times, use
A(t) for the amount of salt (in kg) in the tank
at time t (minutes). We are given that A(0)50.
The rate of change of A could come from salt
being added to the tank (but there is none), or
from salt flowing out of the tank (the solution
flows out at 25 liters per minute, and there are
A(t) kg in 1000 liters, so there are A(t)/40 kg
in 25 liters. So, which of the following is the
differential equation for this problem? A. A'
A/40 B. A' A - 40
C. A' 40 - A D. A' - A/40
E. A' - 40/A
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Answer this...
Now, what is the answer to the problem? (i.e.,
what is A(10) if A -A/40 and A(0) 50 ?) A.
0 B. 40 C. D. E. F. G. 50 ln(2) H.
25
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Connect
Growth and decay
What is the solution of the differential equation
y ky ? How about the initial value problem
y ky , y(0) y0 ? As noted previously,
this differential equation is useful for talking
about radioactive decay, compound interest and
unrestricted population growth.
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One more greatest hits problem
For obvious reasons, the dissecting room of a
medical examiner is kept very cool, at a constant
temperature of 5 degrees C. While doing an
autopsy early one morning, the medical examiner
himself is killed. At 10 am, the examiner's
assistant discovers the body and finds its
temperature to be 23 degrees C, and at noon the
body's temperature is down to 18.5 degrees C.
Assuming that the medical examiner had a normal
temperature of 37 degrees C when he was alive,
when was he murdered? A. 3 am B. 4 am C. 5 am
D. 6 am
E. 7 am F. 8 am G. 9 am
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Geometry of Differential Equations
A differential equation of the form
gives geometric information about the graph of
y(x). It tells us
If the graph of y(x) goes through the point
(x,y), then the slope of the graph at that point
is equal to f(x,y).
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An example
We can draw a picture of this as follows. For the
differential equation
, we have
If the graph goes through (2,3), the slope must
be 1 there. If the graph goes through (0,0), the
slope must be 0 there. If the graph goes through
(-1,-2), the slope must be -1 there.
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Put it on a graph...
The slope of the arrow at any point is equal to y
- x at that point. This kind of picture is called
a "direction field" for the differential equation
dy/dx y - x . We can use this to solve the
differential equation geometrically and recover
the graph of the function.
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The idea is to start
The idea is to start somewhere on the direction
field and simply follow the arrows
This graphical technique is useful for getting
qualitative information about solutions of
differential equations, especially when they
cannot be integrated.
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Here are a couple for you to try...
y ' 2 ( y - y2)
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y ' 3 x sin(2y)
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Numerical methods
Another way to gain insight into solutions of
differential equations is to use numerical
methods for their solution. The simplest
numerical method is called Euler's method.
Euler's method is easy to understand if you
relate it to two things you already know 1. The
left endpoint (rectangle) method for estimating
integrals, and 2. The fundamental theorem of
calculus. Or, you can think of Euler's method
in terms of differentials
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Eulers method
You can algebraically manipulate most first-order
equations, until they are in the form y'(x)
f(x,y) Euler's method then combines the
differential formula with the differential
equation In Euler's method, we simply ignore
the small errors and repeatedly use the resulting
equation with a small value of to
construct a table of values for y(x) (that can
then be graphed, for instance).
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An example...
y ' y - x, y(0) 2 (this is the example we
graphed before). We'll use Dx 0.1 The choice
of Dx is usually dictated by the problem or the
situation. The smaller Dx is the more accurate
the approximated solution will be, but of course
you need to do more work to cover an interval of
a given length. For the first step, we can use
that x0 and y2, therefore y ' 2. Euler's
method then tells us that y(x Dx)
y(x) f(x,y) Dx
y(0.1) 1 (2 - 0) 0.1 1.2
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Continue...
For the second step, we have x 0.1, y 2.2,
therefore y' 2.1. Euler's method then gives
y(0.2) 2.2 2.1(0.1) 2.41 We continue in
this manner and fill in the following table
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Maple...
Maple tells us that the exact solution of the
equation y ' y - x that has y(0) 2 is
y(x) x 1 ex and so we have y(1)
4.718281828. So the Euler method result is
pretty close (within 10). We could do better by
decreasing Dx, but of course then we'd need more
steps to reach x 1. You'll get to try a couple
of these on this week's homework.
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Good night!

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