Exact Equations

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?