Applications of Differential Equations
Modeling a chemical mixture
Suppose that a 10 gallon tank containing a 80% water and 20%
salt solution is being drained at a rate of 2 gallons per minute. If
at the same time a 50% salt solution is being added to the tank at
a rate of 2 gallons per minute, how much salt (in gallons) is in the
tank after 10 minutes? .
Chemical Mixture cont.
Let A(t) be the amount of salt (measured in gallons) in the
mixture at time t (measured in minutes).
dA
A
= −2
+ 1.
dt
10
This can be put into standard form A′ + 15 A = 1. If we solve using
the integrating factor method we find the solution
A(t) = 5 + Ce−t/5 .
Using the initial condition A(0) = 2 we find that C = −3 so
A(t) = 5 − 3e−t/5 .
At t = 10 this implies that there are
A(10) = 5−3e−10/5 = 5−3e−2 ≈ 4.6 gallons of salt in the mixture.
Turning it around a bit...
What if you were asked instead to find out how long it takes to
double the amount of salt in the mixture?
Now you want to know what t value (i.e. when!) you have twice
the amount of salt that you started with.
This requires solving
4 = 5 − 3e−t/5
or
1
1
= e−t/5 ⇒ t = −5 ln( ) ≈ 5.49 minutes.
3
3
Chapter 7 - Functions of Several Variables
7.1: The three-dimensional coordinate system
Learn the (x, y, z) coordinate system
Plotting points in space
Distance formula
Midpoint formula
Equation of a Sphere
Traces of surfaces
(x, y, z) coordinate system and plotting points in space
Practice drawing the (x, y, z) coordinate system and plotting
points in space
z axis
y axis
x axis
Figure: The (x, y, z) coordinate system
(x, y, z) coordinate system and plotting points in space
Make sure you can draw points in all 8 quadrants for both positive
and negative x, y, and z.
z axis
!"#"$%&'& (")"*%&
y axis
x axis
Figure: plot of (x, y, z) = (1, 2, 3)
Distance Formula
If I want to know how far apart 2 points in space are, I can use the
distance formula. In (x, y)−coordinates, recall the formula for the
distance between (x1 , y1 ) and (x2 , y2 ) comes from the pythagorean
formula. We obtain
p
d((x1 , y1 ), (x2 , y2 )) = (x1 − x2 )2 + (y1 − y2 )2 .
For points in (x, y, z)−coordinates we just add the z−component
in the same way so we have
p
d((x1 , y1 , z1 ), (x2 , y2 , z2 )) = (x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 .
Midpoint Formula
If I want to know which point lies at the midpoint of the line
between two points, I just take the average of each of the
components. Namely The point which lies at the midpoint of the
line between (x1 , y1 , z1 ) and (x2 , y2 , z2 ) has coordinates
(xm , ym , zm ) given by
(x1 + x2 ) (y1 + y2 ) (z1 + z2 )
(xm , ym , zm ) =
,
,
.
2
2
2
Equation of Sphere
One important object in three dimensions is a sphere. Recall the
equation for a circle centered at the origin is
x2 + y 2 = r 2 , with r being the radius.
The equation for a sphere centered at the origin is quite similar,
just adding the third component
x2 + y 2 + z 2 = r 2 ,
where once again r is the radius of the sphere. If we want to move
the sphere to be centered at (h, k, j) the formula becomes
(x − h)2 + (y − k)2 + (z − j)2 = r 2 .
Graph x2 + y 2 + z 2 = −2x + 4z + 11
If we complete the square we see that
(x + 1)2 = x2 + 2x + 1
and
and
(y − 0)2 = y 2
(z − 2)2 = z 2 − 4z + 4
so the original equation becomes
(x + 1)2 + (y − 0)2 + (z − 2)2 − 5 = 11
or
(x + 1)2 + (y − 0)2 + (z − 2)2 = 16 = 42
so this is the equation of a sphere centered at (−1, 0, 2) with
radius r = 4.
Trace of Surface
The trace of a surface is a tool to help you graph a surface. The
idea is that you are projecting the surface onto one of the three
planes (x = 0, or y = 0 or z = 0.) In practice if you want the
(x, y) trace of a surface you simply put z = 0 in the formula. And
if you want the (x, z) trace you put y = 0 and if you want the
(y, z) trace you put x = 0. So find the (x, y) trace of the sphere
we just computed
x2 + y 2 + z 2 = −2x + 4z + 11
has (x, y) trace x2 + y 2 = −2x + 11. What does this look like?
(x, y) trace of x2 + y 2 + z 2 = −2x + 4z + 11
x2 + y 2 = −2x + 11 ⇒ (x + 1)2 − 1 + y 2 = 11
which becomes
√
(x + 1)2 + y 2 = ( 12)2 .
This
√ is a circle in the (x, y) plane centered at (−1, 0) with radius
12.