Transcript Mon, Mar 17

Clicker Question 1
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Suppose y  = (x2 – 3x + 2) / x . Then y could
be:
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A. 2x – 3
B. ½ x2 – 3x + 2
C. ½ x2 – 3x + 2 ln(x) + 7
D. ½ x2 – 3 + 2 ln(x)
E. (1/3 x3 – 3/2 x2 + 2 x) / (1/2 x2)
Clicker Question 2
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If f (t) = tan(t), then f (t) could be:
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A. sec2(t)
B. sec(t2)
C. ln(sec(t))
D. ln(tan(t))
E. ln(cos(t))
Clicker Question 3
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Suppose dy / dx = 1 / (1 – x2), then y could
be:
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A. arcsin(x) + 12
B. arctan(x) - 5
C. sin(x) + 43
D. tan(x) – 3.5
E. (1 – x2)-3/2 + e2
Differential Equations (3/17/14)
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A differential equation is an equation which
contains derivatives within it.
More specifically, it is an equation which may
contain an independent variable x (or t)
and/or a dependent variable y (or some other
variable name), but definitely contains a
derivative y ' = dy/dx (or dy/dt).
It may also contain second derivatives y '' ,
etc.
Examples of DE’s
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Every anti-derivative (i.e., indefinite integral)
you have solved (or tried to solve) this
semester is a differential equation!
What is y if y ' = x2 – 3x + 5 ?
What is y if y ' = x / (x2 + 4)
What is y if dy/dt = e0.67t
Note that you also get a “constant of
integration” in the solution.
New types of examples
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The following is a DE of a different type since
it contains the dependent variable:
y ' = .08y
Say in words what this says!
Note that we don’t see the independent
variable at all – let’s call it t .
What is a solution to this equation? And how
can we find it?
The solutions to a DE
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A solution of a given differential equation is
a function y which makes the equation work.
Show that y = Ae0.08t is a solution to the DE
on the previous slide, where A is a constant.
Note that we are using the old tried and true
method for solving equations here called
“guess and check”.
Examples of guess and check for DE’s
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Show that y = 100 – A e –t satisfies the DE
y ' = 100 - y
Show that y = sin(2t) satisfies the DE
d2y / dt 2 = -4y
Show that y = x ln(x) – x satisfies the DE
y ' = ln(x)
Of course one hopes for better methods to
solve equations, but DE’s can be very hard.
Assignment for Wednesday
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Read over these slides (and try to solve the
problems on them), and read Section 9.1.
On page 584, do # 1 – 7 odd.