driffrential eqn and partial fraction.pptx

7/29/2019 driffrential eqn and partial fraction.pptx

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2/11

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


4/11

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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driffrential eqn and partial fraction.pptx

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