exact equations
DESCRIPTION
Exact Equations. Nonlinear Non-separable First Order Equations. Techniques We Know. Nonlinear Separable. Separation of Variables. Integrating Factors. Linear. Non-separable. Nonlinear. Implicit Differentiation. Note:. Implicit Differentiation. Implicit Differentiation. - PowerPoint PPT PresentationTRANSCRIPT
Exact EquationsNonlinear Non-separable First Order Equations
Techniques We KnowNonlinea
rSeparabl
e
Separationof
VariablesLinea
r
Integrating
Factors
Nonlinear
Non-separable
Implicit Differentiation
Note:
Implicit Differentiation
Implicit DifferentiationThis is the general rule:
This comesfrom the
termswith x
This comesfrom the
termswith y
Implicit DifferentiationThis is the general rule:
So to solve problems of the form:
What do we do? Find
:Where:
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
Solving Nonlinear Non-Separable Equations
To solve problems of the form:
Find:Wher
e:
Rewrite as:
Integrate:
Solving Nonlinear Non-Separable Equations
To solve problems of the form:
Find:Wher
e:
Rewrite as:
Integrate:
Finding f(x,y)
Something
Something
(maybe different)
Finding f(x,y)
Nothingin this case
Finding f(x,y)
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…)
Exact EquationsHow do we
know?Remember, order doesn’t matter for partial
derivatives:
So, if:
Then:
Exact EquationsHow do we
know?Remember, order doesn’t matter for partial
derivatives:
So, if:
Then: These are
the same!
Exact EquationsHow do we
know?Remember, order doesn’t matter for partial
derivatives:
So, if:
Then:
Exact Equations
If:
For Equations of the form:
The equation is called Exact
(and we know how to solve)Exampl
e:
Exact Equations
If:
For Equations of the form:
The equation is called Exact
(and we know how to solve)Not an
Example:
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:
Example
Example
Equation is Exact!
Example
Equation is Exact!
Example
Equation is Exact!
General Solution
SummaryNon-
separableNonlinear
Exact
Method of
ExactEquation
s
SummaryNonlinea
rSeparabl
e
Separationof
VariablesLinea
r
Integrating
FactorsNon-separableNonlinear
Exact
Method of
ExactEquation
s
Questions?