Math 2250-4
Wed January 8
Announcement: I modified the first homework assignment slightly - see problems 1.1.26,1.1.28.
1.2 Differential equations of the form y# x = f x
(Also between today's and Monday's notes we will have introduced 1.3: slope fields, and 1.4: separable
differential equations. We will have much more to say about these two sections on Friday and on Monday
next week.)
,
Review from Monday. What were the main ideas we talked about?
,
Use Monday's notes to discuss Newton's law of cooling, and the dead body mystery (Exercise 4)
, Then use Monday's notes to start section 1.2. That discussion continues in today's notes, starting on
the second page.
We will be using the chain rule backwards to solve the Newton's law of cooling differential equation in
Exercise 4 above, just like we did for the exponential growth differential equation on Monday. Here is
justification for the separation of variables algorithm that makes this process easier to carry out for socalled "separable" differential equations:
Definition: A separable first order DE for a function y = y x is one that can be written in the form:
dy
=f x f y .
dx
It's more convenient to rewrite this DE as
1 dy
=f x ,
(as long as f y s 0) .
f y dx
1
Writing g y =
the differential equation reads
f y
dy
g y
=f x .
dx
Solution (math justified): Suppose G y is an antiderivative with respect to y of g y and F x is an
antiderivative of f x with respect to x. By the chain rule, if y x is any solution to the DE on some
interval, then
d
d
G y x = G # y x y# x = g y x y# x = f x =
F x .
dx
dx
Thus
G y x = F x CC .
G y = F x CC
Expresses y implicitly as a function of x . You may be able to use algebra to solve this equation explicitly
for y = y x , and (working the computation backwards) y x will be a solution to the DE. (Even if you
can't algebraically solve for y x , this still yields implicitly defined solutions.)
Solution (differential magic): Treat
dy
as a quotient of differentials dy, dx , and multiply and divide the
dx
DE to "separate" the variables:
dy
f x
=
dx
g y
g y dy = f x dx .
Antidifferentiate each side with respect to its variable (?!)
g y dy = f x dx , i.e.
G y C C1 = F x C C2 0 G y = F x C C .
Agrees!
Section 1.2 continued, from Monday's notes:
Exercise 1: A projectile with very low air resistance is fired almost straight up from the roof of a building
30 meters high, with initial velocity 50 m/s. Its initial horizontal velocity is near zero, but large enough so
that the object lands on the ground rather than the roof.
a) Neglecting friction, how high will the object get above ground?
b) When does the object land ?
Here's another fun example from section 1.2, which also reviews important ideas from Calculus - in
dy
particular we will see how the fact that the slope of a graph y = g x is the derivative
can lead to first
dx
order differential equations.
Exercise 2: (See "A swimmer's Problem" and Example 4 in section 1.2). A swimmer wishes to cross a
river of width w = 2 a , by swimming directly towards the opposite side, with constant transverse velocity
vS . The river velocity is fastest in the middle and is given by an even function of x , for Ka % x % a. The
velocity equal to zero at the river banks. For example, it could be that
x2
vR x = v0 1 K 2 .
a
See the configuration sketches below.
a) Writing the swimmer location at time t as x t , y t , translate the information above into expressions
for x# t and y# t .
b) The parametric curve describing the swimmer's location can also be expressed as the graph of a
function y = y x . Show that y x satisfies the differential equation
v0
dy
x2
=
1K 2 .
dx
vS
a
c) Compute an integral or solve a DE, to figure out how far downstream the swimmer will be when she
reaches the far side of the river.
Exercise 3. Reconstructing the position function from a graph of the velocity function: A car is traveling
along a marked road which we may consider equivalent to an x-axis. Set x 0 = x0 = 0. Recorded below
is a graph of the car velocity v t in miles per minute.
a) Find a formula for x t , for the interval 0 % t % 3 minutes, and graph x = x t .
b) Use the formula for x t to determine how far the car traveled in those three minutes.
c) The distance the car traveled is also computable in terms of an area associated to the velocity graph.
The reason has to do with the Fundamental Theorem of Calculus. Explain the connection, and verify that
this area agrees with your answer from (b).