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?