Exact Equations
Nonlinear Non-separable First Order Equations
Techniques We Know
Nonlinear
Separable
Separation
of Variables
Linear
Integrating
Factors
Non-separable
Nonlinear
Implicit Differentiation
Note:
Implicit Differentiation
Implicit Differentiation
This is the general rule:
This comes
from the terms
with x
This comes
from the terms
with y
Implicit Differentiation
This 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:
Where
:
We’ll talk about
how to do this
in a minute
Find:
Solving Nonlinear
Non-Separable Equations
To solve problems of the form:
Where
:
Find:
Rewrite
as:
Integrate:
Solving Nonlinear
Non-Separable Equations
To solve problems of the form:
Where
:
Find:
Rewrite
as:
Integrate:
Finding f(x,y)
Something
Something
(maybe
different)
Finding f(x,y)
Nothing
in 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 Equations
How do we know?
Remember, order doesn’t matter for partial derivatives:
So,
if:
Then
:
Exact Equations
How do we know?
Remember, order doesn’t matter for partial derivatives:
So,
if:
Then
:
These are
the same!
Exact Equations
How do we know?
Remember, order doesn’t matter for partial derivatives:
So,
if:
Then
:
Exact Equations
For Equations of the
form:
If:
The equation is called Exact (and we know how to solve)
Example:
Exact Equations
For Equations of the
form:
If:
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:
See if Equation is
Exact:
If it is Exact, Find:
Where
:
Rewrite
as:
Integrate:
Example
Example
Equation is Exact!
Example
Equation is Exact!
Example
Equation is Exact!
General
Solution
Summary
Non-separable
Nonlinear
Exact
Method of
Exact
Equations
Summary
Nonlinear
Separable
Separation
of Variables
Linear
Integrating
Factors
Non-separable
Nonlinear
Exact
Method of
Exact
Equations
Questions?