Mathematics 7.pdf

Differential Equations
and Transforms
Mathematics
© 2018 Professional Publications, Inc.

MATHEMATICS
Differential Equations and Transforms
• Differential Equations
• Laplace Transformations
© 2018 Professional Publications, Inc. 165
Lesson Overview

MATHEMATICS
linear differential equation
• includes a function (e.g., y(x)) and its derivatives (e.g., dy/dx)
• order, n , equals the highest derivative
• derivatives aren’t multiplied, squared, involved with trigonometric functions, etc.
• doesn’t include partial derivatives (ordinary)

MATHEMATICS
homogenous differential equation
the response of a system without
external inputs
when f(x) is zero (no input):

MATHEMATICS
homogenous solution
reveals the system’s characteristic
response to initial conditions)
• Ci are constant coefficients
• ri are roots of the system’s
characteristic equation (or system
poles or eigenvalues)

MATHEMATICS
first-order homogeneous linear differential equation
common in engineering models
• frequently,
-a

MATHEMATICS
•
Example: Differential Equations

MATHEMATICS
•
The answer is (B).

MATHEMATICS
second-order homogeneous linear
differential equation
also very common (spring-mass-damper
systems, LRC circuits, and so forth)
solution
• depends on the roots of the
characteristic equation
• determines whether system oscillates

MATHEMATICS
overdamped, a2 > 4b
critically damped, a2 = 4b
underdamped, a2 < 4b

MATHEMATICS
•

MATHEMATICS
•
Find a particular solution. The forcing function is a constant,
so (from the Differential Equations chapter, Table 1) the
particular solution is a constant.
Solution

MATHEMATICS
•
The answer is (C).
Solution (cont.)

MATHEMATICS
Fourier series
• sum of an infinite number of
sinusoidal terms (known as harmonic
terms)
• process of finding the terms is Fourier
analysis

MATHEMATICS
Fourier’s theorem

MATHEMATICS
Fourier coefficients

MATHEMATICS
Laplace Transforms
Laplace transforms
method for solving differential equations,
transient analysis and frequency
response analysis

MATHEMATICS
Laplace Transforms
using Laplace transforms
• put equation in standard form
• take Laplace transform of both sides
• expand terms
• plug in initial conditions y(0) = c,
y(0) = k
• manipulate into a form that has an
inverse transform
• take inverse transform
1
1
0
( ) (0)
( )
n m
n
n n m
n m
m
d f t d f
s F s s
dt dt

 

 
 
 
 

L
     
2
( ) 0 0
 
  
y s y sy y
L L
   
( ) 0
  
y s y y
L L

MATHEMATICS
Example: Laplace Transforms
Solve this linear differential equation by
Laplace transform.
(A)
(B)
(C)
(D)
t
t
e
e 


 3
t
t
e
e 

 3
3
3
9 4
t t
e e
 
 
t
t
e
e 3


   
0 0, 0 2
y y
 
4 3 0
y y y
 
  

MATHEMATICS
Laplace transform.
(A)
(B)
(C)
(D)
Solution
Expand the differential terms.
Substitute initial conditions and
rearrange.
Solve for Y.
Laplace transform.
(A)
(B)
(C)
(D)
t
t
e
e 


 3
t
t
e
e 

 3
3
3
9 4
t t
e e
 
 
t
t
e
e 3


   
0 0, 0 2
y y
 
4 3 0
y y y
 
  
     
 
2
4 3 0
0 0 4 0 3 0
y y y
s Y sy y sY y Y
 
   

     
  
2
4 3 2
3 1 2
s Y sY Y
s s Y
  
  
  
2
3 1
Y
s s

 

MATHEMATICS
Solution (continued)
Separate Y by partial fractions.
For the two sides to be equal,
Solving simultaneous equations gives
  
 
  
 
  
 
  
 
2
3 1 3 1
1 3
3 1 3 1
3
3 1
2 3
A B
s s s s
A s B s
s s s s
A B s A B
s s
A B s A B
 
   
 
 
   
  

 
   
0
3 2
A B
A B
 
 
  
2
3 1
Y
s s

 
1, 1
A B
  
1 1
3 1
Y
s s

 
 

MATHEMATICS
Laplace transform.
(A)
(B)
(C)
(D)
Solution (continued)
From the table of Laplace transform
pairs,
Applying this to both terms in Y,
The answer is (A).
t
e
s





1
3
1 1
3 1
t t
Y e e
s s
 

    
 
Laplace transform.
(A)
(B)
(C)
(D)
t
t
e
e 


 3
t
t
e
e 

 3
3
3
9 4
t t
e e
 
 
t
t
e
e 3


4 3 0
y y y
 
  
   
0 0, 0 2
y y
 

MATHEMATICS
You have reviewed
• basic concepts in areas of
mathematics that are
fundamental to engineering
Learning Objectives

MATHEMATICS
Units
Fundamental Constants
Conversion Factors
Analytic Geometry and Trigonometry
• Straight Lines
• Quadratic Equations
• Conic Sections
• Right Triangles
• Trigonometric Identities
• General Triangles
• Mensuration of Areas
• Mensuration of Volumes
Algebra and Linear Algebra
• Logarithms
• Complex Numbers
• Polar Coordinates
• Matrices
• Solving Simultaneous Linear Equations
• Vectors
• Progressions and Series
Lesson Overview (1 of 2)

MATHEMATICS
Calculus
• Derivatives
• Critical Points
• Partial Derivatives
• Curvature
• Gradient, Divergence, and Curl
• Limits
• Integrals
• Centroids and Moments of Inertia
Differential Equations and Transforms
• Differential Equations
• Laplace Transforms
Lesson Overview (2 of 2)

Mathematics 7.pdf

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