Transcript C.8.4 - Differential Equations

C.9.1 - Differential Equations
Calculus - Santowski
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Lesson Objectives
• 1. Become familiar with a definition of and
terminology involved with differential
equations
• 2. Solve differential equations with and
without initial conditions
• 2. Apply differential equations in a variety
of real world applications
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(A) Definitions
• A differential equation is any equation which
contains derivatives
dy
 x sin y
• Ex:
dx
• Ex: Newton’s second law of motion can also be
rewritten as a differential equation:
• F = ma which we can rewrite as
dv
d 2s
Fm
or F  m 2
dt
dt
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(A) Definitions
• The order of a differential equation refers
to the largest derivative present in the DE
• A solution to a differential equation on a
given interval is any function that satisfies
the differential equation in the given
interval
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(B) Example

3
2
• Show that y(x)  x
is a solution to the
2
4x
y 12xy  3y  0
second order DE
for x > 0
• Why
 is the restriction x > 0 necessary?

• Are the equations listed below also
solutions??
1
3
3
y(x)  x 2 or y(x)  9x
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
2
or y(x)  9x
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
2
 7x

1
2
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(C) Initial Conditions
• As we saw in the last example, a DE will
have multiple solutions (in terms of many
functions that satisfy the original DE)
• So to be able to identify or to solve for a
specific solution, we can given a condition
(or a set of conditions) that will allow us to
identify the one single function that satisfies
the DE
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(D) Example - IVP

3
2
• Show that y(x)  x is a solution to the
2
4x
y 12xy  3y  0
second order DE
given the initial conditions of

1
3
y(4) 
and y (4)  
 8
64

3
2
• What about the equation y(x)  9x  7x
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
1
2
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(D) Examples
3 c
• Ex 1. Show that y(t)  4  t 2
is the
general solution to the DE 2ty  4 y  3
 is the actual solution to the IVP
• Ex 2. What
2ty  4 y  
3 and y(1)  4

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(E) Further Examples
• Ex 1. What is the solution to the first order
differential equation (FODE)
dy
 2x
dx
• Sketch your solution on a graph.

• Then solve the IVP given the initial condition of
(-1,1)
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(E) Further Examples
• Ex 2. Solve the IVP
dy
1
 sin x 
and e,1 cos(e)
dx
x
• How would you verify your solution?

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(F) Motion Examples
• If a body moves from rest so that its
velocity is proportional to the time elapsed,
show that v2 = 2as
where s = displacement, t is time and v is
velocity and a is a constant (representing
….?)
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(F) Motion Examples
• A particle is accelerated in a line so that its
velocity in m/s is equal to the square root of
the time elapsed, measured in seconds. How
far does the particle travel in the first hour?
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(F) Motion Examples
• A stone is tossed upward with a velocity of
8 m/s from the edge of a cliff 63 m high.
How long will it take for the stone to hit the
ground at the foot of the cliff?
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(F) In Class Worksheets
• We will work through selected questions
from handouts from several other Calculus
textbooks
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(G) DE on TI-89
Qui ckTi me™ and a
TIFF (Uncompressed) decompressor
are needed to see this pictur e.
Qui ckTi me™ and a
TIFF (Uncompressed) decompressor
are needed to see this pictur e.
Qui ckTi me™ and a
TIFF (Uncompressed) decompressor
are needed to see this pictur e.
Qui ckTi me™ and a
TIFF (Uncompressed) decompressor
are needed to see this pictur e.
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(G) DE on TI-89
• So the last slides have
shown how to solve
DE on the TI-89 as
well as preparing
graphs (which we will
explore in more detail
in a future lesson)
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Qui ckTi me™ and a
TIFF (Uncompressed) decompressor
are needed to see this pictur e.
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