driffrential eqn and partial fraction.pptx
TRANSCRIPT
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Differential Equations
4 2 2 3sin , ' 2 0, 0y x y y xy x y y x
Definition
A differential equation is an equation involving
derivatives of an unknown function and possiblythe function itself as well as the independentvariable.
Example
Definition The order of a differential equation is the highestorder of the derivatives of the unknown functionappearing in the equation
1st order equations 2nd order equation
sin cosy x y x C
Examples 2 3
1 1 26 e 3 e e
x x xy x y x C y x C x C
In the simplest cases, equations may be solved by directintegration.
Observe that the set of solutions to the above 1st orderequation has 1 parameter, while the solutions to the above 2ndorder equation depend on two parameters.
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Separable Differential Equations
A separable differential equation can be expressed as the product of a
function ofx and a function ofy.
dy
g x h ydx
Example:
22
dyxy
dx
Multiply both sides by dx and divide both sides byy2 to separate the variables. (Assumey2 is neverzero.)
22dy x dx
y
22y dy x dx
0h y
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A separable differential equation can be expressed as the product of a
function ofx and a function ofy.
dy
g x h ydx
Example:
22
dyxy
dx
22dy x dx
y
22y dy x dx
22y dy x dx
1 2
1 2y C x C
21 x Cy
2
1y
x C
2
1y
x C
0h y
Combined
constants of
integration
Separable Differential Equations
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Partial Fraction
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Partial Fraction Theory
Integration theory, algebraic manipulations and
Laplace theory all use partial fraction theory, whichapplies to polynomial fractions
a0 + a1s + + ansn
b0 + b1s + + bmsm
where the degree of the numerator is less than thedegree of the denominator.
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2
78
2
xx
xFind the partial fraction decomposition for:
As we saw in the previous slide the denominator factors as (x +2)(x 1). We want to find numbers A and B so that:
122
78
2
x
B
x
A
xx
x
The bad news is that we have to do this without peeking at the previous slide tosee the answer. What do you think will be our first move?
Congratulations if you chose multiplying both sides of the equation by the LCD.The good news is that, since we are solving an equation, we can get rid offractions by multiplying both sides by the LCD.
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So we multiply both sides of the equation by (x + 2)(x 1).
If the left side and the right sideare going to be equal then:
A+B has to be 8 and
-A+2B has to be 7.
BAxBAxBBxAAxx
xBxAx
xxx
Bxx
x
A
xx
xxx
xxx
B
x
A
xx
xxx
278
278
2178
121
12212
7812
12122
7812
2
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A + B = 8
-A + 2B = 7
This gives us two equations in two unknowns. We can add the twoequations and finish it off with back substitution.
3B = 15
B = 5
If B = 5 and A + B = 8 then A = 3.
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Summary of Partial Fraction Decomposition
When Denominator Factors Into Linear
Factors (Factors of first degree)
Factor the denominator
Set fraction equal to sum of fractions with each factoras a denominator using A, B, etc. for numerators
Clear equation of fractions
Use convenient xmethod to find A, B, etc.
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