Objective: Find exact and approximate solutions of equations that contain decimals.
Exact Equations
description
Transcript of Exact Equations
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Exact EquationsNonlinear Non-separable First Order Equations
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Techniques We KnowNonlinea
rSeparabl
e
Separationof
VariablesLinea
r
Integrating
Factors
Nonlinear
Non-separable
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Implicit Differentiation
Note:
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Implicit Differentiation
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Implicit DifferentiationThis is the general rule:
This comesfrom the
termswith x
This comesfrom the
termswith y
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Implicit DifferentiationThis is the general rule:
So to solve problems of the form:
What do we do? Find
:Where:
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Solving Nonlinear Non-Separable Equations
To solve problems of the form:
Find:Wher
e:
We’ll talk about
how to do this
in a minute
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Solving Nonlinear Non-Separable Equations
To solve problems of the form:
Find:Wher
e:
Rewrite as:
Integrate:
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Solving Nonlinear Non-Separable Equations
To solve problems of the form:
Find:Wher
e:
Rewrite as:
Integrate:
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Finding f(x,y)
Something
Something
(maybe different)
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Finding f(x,y)
Nothingin this case
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Finding f(x,y)
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The catch…(Because there’s always a
catch)
Try to find:Where:
(Hint: You’re going to be trying for a really long time…)
(Hint: Because it can’t be done…)
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Exact EquationsHow do we
know?Remember, order doesn’t matter for partial
derivatives:
So, if:
Then:
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Exact EquationsHow do we
know?Remember, order doesn’t matter for partial
derivatives:
So, if:
Then: These are
the same!
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Exact EquationsHow do we
know?Remember, order doesn’t matter for partial
derivatives:
So, if:
Then:
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Exact Equations
If:
For Equations of the form:
The equation is called Exact
(and we know how to solve)Exampl
e:
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Exact Equations
If:
For Equations of the form:
The equation is called Exact
(and we know how to solve)Not an
Example:
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Solving Nonlinear Non-Separable Equations
To solve problems of the form:
If it is Exact, Find:Wher
e: Rewrite as:
Integrate:
See if Equation is Exact:
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Example
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Example
Equation is Exact!
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Example
Equation is Exact!
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Example
Equation is Exact!
General Solution
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SummaryNon-
separableNonlinear
Exact
Method of
ExactEquation
s
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SummaryNonlinea
rSeparabl
e
Separationof
VariablesLinea
r
Integrating
FactorsNon-separableNonlinear
Exact
Method of
ExactEquation
s
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Questions?