Differential Equations Corel

8/19/2019 Differential Equations Corel

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Differential EquationsPrinciples and Applications


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A. ESSENTIAL CONCEPTS & FORMULAS

1. Definition: Differential Equation

2. Classifications of D.E.

- ODE

- PDE

- Order of D.E.

- Linearity of D.E.


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1. Differential Equations

- contains the derivative (differentials) of one or moredependent variable with respect to one or more dependentvariables.

examples: acceleration of a falling object =

= 9.81


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2. CLASSIFICATIONS OF D.E.

2.1. ACCORDING TO TYPE

A. ODE(Ordinary Differential Equations)

- contains only ordinary derivatives of one or more

functions with respect to a single independent variable


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2.1. ACCORDING TO TYPE

B. PDE(Partial Differential Equations)

- An equation involving partial derivatives of one ormore unknown functions of two or more independentvariables


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2.2. ACCORDING TO ORDER

- the order of D.E. is the HIGHEST derivative in the equatio


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2.3. ACCORDING TO LINEARITY

- An nth order D.E. Is said to be linear if the function is line

if the function is linear in y, y’, y’’,..., y(n)

Examples of Linear D.E.

Examples of Nonlinear D.E.


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Classify the following D.E. according to:(a) types (b) order & degree (c) linearity

1.

= 1

2. ′ + ′ =

3. ′′′ + 8′′ = + 1

4.

^

6

+ 4 = 0

5. ′ =

6.

4

= 2

7. ′′ + 8′ + = sin


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B. SOLUTIONS TO DIFFERENTIAL EQUATIONS

1. General Solution

2. Particular Solution

3. Initial-Value Problem

4. Solutions to 1st Order D.E. using

- Integration

- Separation of variables- Homogeneous

- Linear equation

- Bernoulli equation

- Exact equation

- Non-exact D.E.


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1. General Solution

- A solution to D.E. which contains one or more arbitraryconstants of integration ‘C’

= ()+ C

As an example, obtain the equation of the velocity of an objectdropped from rest at ant time ‘t’. a = -9.81 m/s2.

= 9.81 + (Answer)


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2. Particular Solution

- A single solution to D.E. given an additional information tsolve for the arbitrary constants of integration ‘C’

As an example, if the initial velocity dropped is -20m/s, obtain thequation of the velocity of an object dropped from rest at ant tim‘t’. a = -9.81 m/s2.

= 9.81 20 (Answer)


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3. Initial Value Problem

As an example, solve the D.E.

= 1, given the boundary

condition, y = 5 when x = 0.

y =

+ 5 (Answer)


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3. SOLUTIONS TO 1ST ORDER D.E.

Example: Obtain the general solution

(answer)


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Example: Obtain the general solution

(answer)


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(answer)


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(answer


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(answer


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(answer


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(answer


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G. NON EXACT


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Sample problemsSolve the following differential equations.

1.

2.

3.

4.

5.


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APPLICATIONS OF 1st ORDER D.E.

1. GROWTH and DECAY PROBLEMS1. HALF-LIFE PROBLEMS

2. TEMPERATUE PROBLEMS1. NEWTON’S LAW OF COOLING

3. FREE FALLING BODIES

4. MIXTURES


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GROWTH and DECAY PROBLEMS

“The rate of change of a substance is proportional to the amount ofsubstance”

=

Solution: N =

where: N – amount of substance at any time t

No – initial amount

t – time

– proportionality constant


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1.DECAY PROBLEMIf 6% of a substance decomposes in 50 years

(a) What percent will be present at the end of 400 years?

(b) in 800 years?

(c) what is the half-life of the substance

ANSWERS (a) 61% Table x(time) y(amount)

(b) 37% 0 100%

(c) 560 years 50 94%

400 ??? (ANS. 6


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2.GROWTH PROBLEMIf a population of a town in Luzon doubled in past 25 years

and the present population is 300,000, when will the town have apopulation of 800,000?

ANSWERS 60.37 years Table x(time) y(population)0 150,000

25 300,000

(ANS. 60.37 yrs) ??? 800,000


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TEMPERATURE PROBLEMS (Newton’s Law of Cooling)

“The rate of change of the temperature of a body is proportional to ttemperature difference betwen the body and its surrounding medium

= ( )

Solution: =

where: T – temp of a body at any time t

o – initial temperature at time t=0

t – time



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TEMPERATURE PROBLEM

1. NEWTON’S LAW OF COOLINGA cup of hot chocolate whose temperature is 200F is poure

in a room whose temperature is 70F. Two minutes later thetemperature of the chocolate is 180F. How long after the chocolais poured does it reach a temperature of 155F?

ANSWERS 5.08 mins. Table x(time) y(population)

0 200 – 70

2 180 – 70

(ANS. 5.08 mins.) ??? 155 - 70


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FREE – FALLING BODIES (Newton’s 2nd Law of Motion)

“The net force acting on a body is equal to the time rate of chanof the momentum of the momentum of the body, for constant

mass”

= =

Example: A rock weighs 32lb is dropped from rest from the edge ocliff (a) find the velocity of the rock at time t if the air resistanceequivalent to the instantaneous velocity v

(b) What is the velocity of the rock at t= 2sec?

ANSWERS (a) = 32 (1 −

) fps (b) 27.66 fps


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MIXTURE PROBLEMS

“The Amount of solution in the tank at anytime t is equals to thnet rate”

=

where: – amount of substance at any time t o – initial amount

- rate inflow

– rate of discharge

t – time



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MIXTURE PROBLEMS

A 100-liter tank contains a 25% dye solution. A 40% dye solution iallowed to enter the tank at the rate of 10 liters/minute and theresulting mixture is removed from the tank at the same rate.Derive an expression for the amount of dye in the tank as a function of time t.

Solution:

= (0.40)(10) (

)(10)

Answer : = 40 15−


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SECOND-ORDER DIFFERENTIAL EQUATION

1. 2ND ORDER D.E.

2. SOLUTIONS TO LINEAR HOMOGENEOUS D.E.

3. SOLUTIONS TO LINEAR NON-HOMOGENEOUS D.E.


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SECOND-ORDER DIFFERENTIAL EQUATION

2nd ORDER DIFFERENTIAL EQUATIONS

(a)

+ 3

+ 4 = 3

(b) 6′′ + 4′ = 2


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HOMOGENEOUS LINEAR D.E.

Give the 2nd order D.E. ′′ + ′ + = 0

substituting m = D + + = 0 characteristic equatio

= 1 = 2 roots

Case 1: if roots are distinct = 1 = 2

then y ℎ = 1

+ 2

Case 2: if roots are equal = 1 = 2

then y ℎ = 1 + 2

Case 3: if roots are complex nos = ±

then y ℎ = 1 + 2


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HOMOGENEOUS LINEAR D.E.

EXAMPLES: Find the general solution

1. 2′′ 5′ 3 = 0

2. ′′

10′

+ 25 = 0

3. ′′ + 4′ + 7 = 0


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NON-HOMOGENEOUS LINEAR D.E.

NON-HOMOGENEOUS FUNCTION IN THE FORM

′′ + ′ + = ℎ:

UNDETERMINED COEFFICIENT SET

Case 1: if ℎ

then y = {, −, −, −, … }

Case 2: if ℎ

then y = {}

Case 3: if ℎ sin bx or cos bx

theny = +


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TOTAL DIFFERENTIAL EQUATION

• TOTAL D.E.

y = ℎ + ()

1. Solve the total d.e. of ′′ 9 = 4−

(answer) = 1 + 2 −

−


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BREAK TIME


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Dynamics


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TOPIC OUTLINE

1. CONSTANT VELOCITY

2. CONSTANT ACCELARATION

3. VARIABLE ACCELERATION

4. PROJECTILE MOTION

5. RELATIVE MOTION


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DIFFERENTIAL EQUATIONS OF MOTION


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A. CONSTANT ACCELERATION


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CURVILINEAR MOTION


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PROJECTILE MOTION


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RELATIVE MOTION

POSITION

VELOCITY

ACCELERATION


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GOOD LUCK FOR YOUR EXAMS!

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Differential Equations Corel

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