Exact EquationsNonlinear Non-separable First Order Equations

Techniques We Knowm (y)y0 =n (x)

y0 + p (t)y = q (t)

Nonlinear Separable

Separation of Variables

Linear Integrating Factors

M (x, y) +N (x, y)y0 = 0

Nonlinear

Non-separable

Implicit Differentiation5x2

y

3 + 7x3� �0

= 10xy3 21x215x2y

2+ +y0

@

@x

�5x2

y

3 + 7x3�= 10xy3+ 21x2

@

@y

�5x2

y

3 + 7x3�= 15x2

y

2

Note:

Implicit Differentiation5x2

y

3 + 7x3� �0

= 10xy3 21x2 15x2y

2+ + y0

@

@x

�5x2

y

3 + 7x3�= 10xy3+ 21x2

@

@y

�5x2

y

3 + 7x3�= 15x2

y

2

f (x, y)0 = f

x

(x, y) +fy (x, y)y0

Implicit Differentiation

f (x, y)0 =f

x

(x, y)+fy (x, y)y0

This is the general rule:

This comes from the terms

with x

This comes from the terms

with y

Implicit DifferentiationThis is the general rule:

So to solve problems of the form:

M (x, y) +N (x, y)y0 = 0

f (x, y)0 =f

x

(x, y)+fy (x, y)y0

What do we do?Find: f (x, y)

Where:f

x

(x, y)=M (x, y)

fy (x, y) = N (x, y)

Solving Nonlinear Non-Separable Equations

To solve problems of the form:M (x, y) +N (x, y)y0 = 0

Find: f (x, y)

Where:f

x

(x, y)=M (x, y)

fy (x, y) = N (x, y)

We’ll talk about how to do this

in a minute

Solving Nonlinear Non-Separable Equations

To solve problems of the form:M (x, y) +N (x, y)y0 = 0

Find: f (x, y)

Where:f

x

(x, y)=M (x, y)

fy (x, y) = N (x, y)

Rewrite as:f (x, y)0 = 0

Integrate:Zf (x, y)0 dt =

Z0dt

Solving Nonlinear Non-Separable Equations

To solve problems of the form:M (x, y) +N (x, y)y0 = 0

Find: f (x, y)

Where:f

x

(x, y)=M (x, y)

fy (x, y) = N (x, y)

Rewrite as:f (x, y)0 = 0

Integrate:f (x, y) = C

Finding f(x,y)

f

x

(x, y) =M (x, y)

fy (x, y) =N (x, y)

Zdx

Zdx

= +g (y)Something

Zdy

Zdy =

Something (maybe

different)+h (x)

Finding f(x,y)

= +g (y)

Zdy = +h (x)

10xy3 21x2 15x2y

2+ + y0 = 0

Z10xy3 + 21x2

dx 5x2y

3 + 7x3

15x2y

2 5x2y

3 Nothing in this case

Finding f(x,y)

= +g (y)

Zdy = +h (x)

10xy3 21x2 15x2y

2+ + y0 = 0

Z10xy3 + 21x2

dx 5x2y

3 + 7x3

15x2y

2 5x2y

3

f (x, y) = 5x2y

3 + 7x3

The catch…(Because there’s always a catch)

3y + 2xy0 = 0

Try to find: f (x, y)

Where:f

x

(x, y)=

fy (x, y) =

3y

2x

(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:fxy

= fyx

So, if:f

x

(x, y)=M (x, y)

fy (x, y)=N (x, y)

Then:f

xy

(x, y)=My (x, y)

N

x

(x, y) = f

yx

(x, y)

Exact EquationsHow do we know?

Remember, order doesn’t matter for partial derivatives:fxy

= fyx

So, if:f

x

(x, y)=M (x, y)

fy (x, y)=N (x, y)

Then:=

My (x, y)

N

x

(x, y) =

f

xy

(x, y)

f

yx

(x, y)These are the same!

Exact EquationsHow do we know?

Remember, order doesn’t matter for partial derivatives:fxy

= fyx

So, if:f

x

(x, y)=M (x, y)

fy (x, y)=N (x, y)

Then:My (x, y) N

x

(x, y)=

Exact Equations

If:My (x, y) N

x

(x, y)=

For Equations of the form:M (x, y) +N (x, y)y0 = 0

The equation is called Exact (and we know how to solve)Example: 10xy3 21x2 15x2

y

2+ + y0 = 0

@

@y10xy3 + 21x2 = 30xy2

@

@x

15x2y

2= 30xy2

Exact Equations

If:My (x, y) N

x

(x, y)=

For Equations of the form:M (x, y) +N (x, y)y0 = 0

The equation is called Exact (and we know how to solve)Not an Example: 3y + 2xy0 = 0

@

@y=

@

@x

=

3y

2x

3

2

Solving Nonlinear Non-Separable Equations

To solve problems of the form:M (x, y) +N (x, y)y0 = 0

If it is Exact, Find: f (x, y)

Where:f

x

(x, y)=M (x, y) fy (x, y) = N (x, y)

Rewrite as:f (x, y)0 = 0

Integrate:f (x, y) = C

See if Equation is Exact: My (x, y) N

x

(x, y)=

Exampley

0 = �2x+ y

2

2xy

y0 =2xy2x+ y

2+ 0

y0 =+ 02xy2x+ y

2

M (x, y) N (x, y)

My (x, y) =�2x+ y

2�y= 2y

N

x

(x, y)= = 2y(2xy)x

Exampley

0 = �2x+ y

2

2xyy0 =+ 02xy2x+ y

2

�2x+ y

2�y= (2xy)

x

Equation is Exact!Z

2x+ y

2dx

=x

2 + xy

2+g (y)

Z2xydy = xy

2+h (x)

f (x, y) = x

2 + xy

2

Exampley

0 = �2x+ y

2

2xyy0 =+ 02xy2x+ y

2

�2x+ y

2�y= (2xy)

x

Equation is Exact!Z

2x+ y

2dx

=x

2 + xy

2+g (y)

Z2xydy = xy

2+h (x)

f (x, y) = x

2 + xy

2

�x

2 + xy

2�0= 0Z �

x

2 + xy

2�0dx =

Z0dx

Exampley

0 = �2x+ y

2

2xyy0 =+ 02xy2x+ y

2

�2x+ y

2�y= (2xy)

x

Equation is Exact!Z

2x+ y

2dx

=x

2 + xy

2+g (y)

Z2xydy = xy

2+h (x)

f (x, y) = x

2 + xy

2

�x

2 + xy

2�0= 0

x

2 + xy

2 = C

General Solution

Summary

M (x, y) +N (x, y)y0 = 0 Non-separable Nonlinear

Exact

Method of Exact

EquationsMy (x, y) N

x

(x, y)=

Summarym (y)y0 =n (x)

y0 + p (t)y = q (t)

Nonlinear Separable

Separation of Variables

Linear Integrating Factors

M (x, y) +N (x, y)y0 = 0 Non-separable Nonlinear

Exact

Method of Exact

EquationsMy (x, y) N

x

(x, y)=

Questions?