![Page 1: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/1.jpg)
Exact EquationsNonlinear Non-separable First Order Equations
![Page 2: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/2.jpg)
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
![Page 3: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/3.jpg)
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:
![Page 4: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/4.jpg)
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
![Page 5: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/5.jpg)
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
![Page 6: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/6.jpg)
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)
![Page 7: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/7.jpg)
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
![Page 8: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/8.jpg)
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
![Page 9: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/9.jpg)
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
![Page 10: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/10.jpg)
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)
![Page 11: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/11.jpg)
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
![Page 12: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/12.jpg)
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
![Page 13: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/13.jpg)
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…)
![Page 14: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/14.jpg)
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)
![Page 15: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/15.jpg)
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!
![Page 16: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/16.jpg)
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)=
![Page 17: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/17.jpg)
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
![Page 18: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/18.jpg)
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
![Page 19: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/19.jpg)
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)=
![Page 20: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/20.jpg)
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
![Page 21: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/21.jpg)
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
![Page 22: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/22.jpg)
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
![Page 23: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/23.jpg)
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
![Page 24: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/24.jpg)
Summary
M (x, y) +N (x, y)y0 = 0 Non-separable Nonlinear
Exact
Method of Exact
EquationsMy (x, y) N
x
(x, y)=
![Page 25: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/25.jpg)
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)=
![Page 26: Exact Equations - University of California, San Diegoj9hall/teaching/Math20D/Slides/Week3/Lesson5... · Exact Equations If: M y (x,y) = Nx (x,y) For Equations of the form: M (x,y)](https://reader036.vdocument.in/reader036/viewer/2022062909/5b08a8457f8b9a5f6d8cd0a6/html5/thumbnails/26.jpg)
Questions?