CHAPTER 1: FIRST ORDER ORDINARY DIFFERENTIAL EQUATION

Transcription

1 Classification by type - Ordinary Differential Equations (ODE) Contains one or more dependent variables with respect to one independent variable is the dependent variable while is the independent variable is the dependent variable while is the independent variable Dependent Variable: Independent Variable: - Partial Differential Equations (PDE) involve one or more dependent variables and two or more independent variables Can you determine which one is the DEPENDENT VARIABLE and which one is the INDEPENDENT VARIABLES from the following equations??? Dependent Variable: Independent Variable: Dependent Variable: Independent Variable: 1

2 Classification by order / degree - Order of Differential Equation Determined by the highest derivative - Degree of Differential Equation Exponent of the highest derivative Examples: a) Order : Degree: b) Order : Degree: c) Order : Degree: d) Order : Degree: Classification as linear / nonlinear - Linear Differential Equations Dependent variables and their derivative are of degree 1 Each coefficient depends only on the independent variable A DE is linear if it has the form Examples: 1) 2) 3) 2

3 - Nonlinear Differential Equations Dependent variables and their derivatives are not of degree 1 Examples: 1) Order : Degree: 2) Order : Degree: 3) Order : Degree: Initial & Boundary Value Problems Initial conditions : will be given on specified given point Boundary conditions : will be given on some points Examples : 1) Initial condition 2) Boundary condition Initial Value Problems (IVP) Initial Conditions: Boundary Value Problems (BVP) Boundary Conditions: 3

4 Solution of a Differential Equation - General Solutions Solution with arbitrary constant depending on the order of the equation - Particular Solutions Solution that satisfies given boundary or initial conditions Example: (1) Show that the above equation is a solution of the following DE (2) Solutions: Insert (1) and (4) into (2) Proven that is the solution for the given DE. EXERCISE: Show that is the solution of the following DE 4

5 Forming a Differential Equation Example 1: Find the differential equation for Solution: Try to eliminate A by, a) Divide (1) with : b) (2)+(3) : Example 2: Form a suitable DE using Solutions: 5

6 Exercise: Form a suitable Differential Equation using Hints: 1. Since there are two constants in the general solution, has to be differentiated twice. 2. Try to eliminate constant A and B. 6

7 1.2 First Order Ordinary Differential Equations (ODE) Types of first order ODE - Separable equation - Homogenous equation - Exact equation - Linear equation - Bernoulli Equation Separable Equation How to identify? Suppose Hence this become a SEPARABLE EQUATION if it can be written as Method of Solution : integrate both sides of equation 7

8 Example 1: Solve the initial value problem Solution: i) Separate the functions ii) Integrate both sides iii) Use the initial condition given, iv) Final answer Note: Some DE may not appear separable initially but through appropriate substitutions, the DE can be separable. 8

9 Example 2: Show that the DE substitution can be reduced to a separable equation by using. Then obtain the solution for the original DE. Solutions: i) Differentiate both sides of the substitution wrt ii) Insert (2) and (1) into the DE iii) Write (3) into separable form iv) Integrate the separable equation Final answer : 9

10 Exercise: 1) Solve the following equations a. b. c. d. (Hint: Use ) 2) Using substitution, convert to a separable equation. Hence solve the original equation Homogenous Equation How to identify? Suppose, is homogenous if for every real value of Method of Solution : i) Determine whether the equation homogenous or not ii) Use substitution and in the original DE iii) Separate the variable and iv) Integrate both sides v) Use initial condition (if given) to find the constant value Separable equation method 10

11 Example 1: Determine whether the DE is homogenous or not a) b) Solutions: a) b) 11

12 Example 2: Solve the homogenous equation Solutions: i) Rearrange the DE ii) Test for homogeneity iii) Substitute and into (1) iv) Solve the problem using the separable equation method Final answer : Note: Non-homogenous can be reduced to a homogenous equation by using the right substitution. 12

13 Example 3: Find the solution for this non-homogenous equation by using the following substitutions Solutions: i) Differentiate (2) and (3) (1) (2),(3) and substitute them into (1), ii) Test for homogeneity, iii) Use the substitutions and iv) Use the separable equation method to solve the problem Ans: 13

14 1.2.3 Exact Equation How to identify? Suppose, Therefore the first order DE is given by Condition for an exact equation. Method of Solution (Method 1): i) Write the DE in the form And test for the exactness ii) If the DE is exact, then (1), (2) To find, integrate (1) wrt to get From (1) : (3) 14

15 iii) To determine, differentiate (3) wrt to get iv) Integrate to get v) Replace into (3). If there is any initial conditions given, substitute the condition into the solution. vi) Write down the solution in the form, where A is a constant Method of Solution (Method 2): i) Write the DE in the form And test for the exactness ii) If the DE is exact, then (1), (2) iii) To find from, integrate (1) wrt to get (3) iv) To find from, integrate (2) wrt to get (4) v) Compare and to get value for and. vii) Replace into (3) OR into (4). viii) If there are any initial conditions given, substitute the conditions into the solution. ix) Write down the solution in the form, where A is a constant 15

16 Example 1: Solve Solution (Method 1): i) Check the exactness ii) Find To find, integrate either (1) or (2), let s say we take (1) iii) Now we differentiate (3) wrt to compare with Now, let s compare (4) with (2) iv) Find 16

17 v) Now that we found, our new should looks like this vi) Write the solution in the form Final answer: Exercise : Try to solve Example 1 by using Method 2 Note: Some non-exact equation can be turned into exact equation by multiplying it with an integrating factor. 17

18 Example 2: Show that the following equation is not exact. By using integrating factor, equation. Solution: i) Show that it is not exact, solve the ii) Multiply into the DE to make the equation exact iii) Check the exactness again iv) Find To find, integrate either (1) or (2), let s say we take (2) 18

19 v) Find Now, let s compare (4) with (1) vi) Write Final answer :, where Exercises : 1. Try solving Example 2 by using method Determine whether the following equation is exact. If it is, then solve it. a. b. c. 3. Given the differential equation i. Show that the differential equation is exact. Hence, solve the differential equation by the method of exact equation. ii. Show that the differential equation is homogeneous. Hence, solve the differential equation by the method of homogeneous equation. Check the answer with 3i. 19

20 1.2.4 Linear First Order Differential Equation How to identify? The general form of the first order linear DE is given by When the above equation is divided by, ( 1 ) Where Method of Solution : and Must be NOTE: here!! i) Determine the value of dan such that the coefficient of is 1. ii) Calculate the integrating factor, iii) Write the equation in the form of iv) The general solution is given by 20

21 Example 1: Solve this first order DE Solution: i) Determine and ii) Find integrating factor, iii) Write down the equation iv) Final answer Note: Non-linear DE can be converted into linear DE by using the right substitution. 21

22 Example 2: Using, convert the following non-linear DE into linear DE. Solve the linear equation. Solutions: i) Differentiate to get and replace into the non-linear equation. ii) Change the equation into the general form of linear equation & determine and iii) Find the integrating factor, iv) Find 22

23 v) Use the initial condition given, Equations of the form How to identify? When the DE is in the form ( 1 ) use substitution ( 2 ) to turn the DE into a separable equation Method of Solution : i) Differentiate ( 2 ) wrt ( to get ) ( 3 ) ii) Replace ( 3 ) into ( 1 ) iii) Solve using the separable equation solution 23

24 Example 1: Solve Solution: i) Write the equation as a function of ii) Let and differentiate it to get iii) Replace ( 2 ) into ( 1 ) iv) Solve using the separable equation solution 24

25 1.2.6 Bernoulli Equation How to identify? The general form of the Bernoulli equation is given by ( 1 ) where To reduce the equation to a linear equation, use substitution ( 2 ) Method of Solution : iv) Divide ( 1 ) with ( 3 ) v) Differentiate ( 2 ) wrt ( to get ) ( 4 ) vi) Replace ( 4 ) into ( 3 ) vii) Solve using the linear equation solution Find integrating factor, Solve 25

26 Example 1: Solve ( 1 ) Solutions: i) Determine ii) Divide ( 1 ) with iii) Using substitution, iv) Replace ( 3 ) into ( 2 ) and write into linear equation form v) Find the integrating factor vi) Solve the problem vii) Since 26

27 Exercise: Answer: Given the differential equation, 2 x4y dy (4 x3y2 x3) dx 0. Show that the equation is exact. Hence solve it. 7. The equation in Question 6 can be rewritten as a Bernoulli equation, 2x dy 4 y 1. dx y By using the substitution z y 2, solve this equation. Check the answer with Question 6. 27

28 1.3 Applications of the First Order ODE The Newton s Law of Cooling The Newton s Law of Cooling is given by the following equation Where is a constant of proportionality is the constant temperature of the surrounding medium General Solution Do you know what type of DE is this? Q1: Find the solution for It is a separable equation. Therefore Q2: Find when Q3: Find. Given that 28

29 Example 1: According to Newton s Law of Cooling, the rate of change of the temperature satisfies Where is the ambient temperature, is a constant and is time in minutes. When object is placed in room with temperature C, it was found that the temperature of the object drops from 9 C to C in 30 minutes. Then determine the temperature of an object after 20 minutes. Solution: i) Determine all the information given. Room temperature = C When When Question: Temperature after 20 minutes, ii) Find the solution for iii) Use the conditions given to find and iv) 29

30 Exercise: 1. A pitcher of buttermilk initially at 25:C is to be cooled by setting it on the front porch, where the temperature is. Suppose that the temperature of the buttermilk has dropped to after 20 minutes. When will it be at? 2. Just before midday the body of an apparent homicide victim is found in a room that is kept at a constant temperature of 70:F. At 12 noon, the temperature of the body is 80:F and at 1pm it is 75:F. Assume that the temperature of the body at the time of death was 98.6:F and that it has cooled in accord with Newton s Law. What was the time of death? 30

31 Natural Growth and Decay The differential equation what type of DE Where is this? is a constant is the size of population / number of dollars / amount of radioactive The problems: 1. Population Growth 2. Compound Interest 3. Radioactive Decay 4. Drug Elimination Do you know Example 1: A certain city had a population of in 1960 and a population of in Assume that its population will continue to grow exponentially at a constant rate. What populations can its city planners expect in the year 2000? Solution: 1) Extract the information 2) Solve the DE Separate the equation and integrate 31

32 3) Use the initial & boundary conditions Exercise: 1) (Compounded Interest) Upon the birth of their first child, a couple deposited RM5000 in an account that pays 8% interest compounded continuously. The interest payments are allowed to accumulate. How much will the account contain on the child s eighteenth birthday? (ANS: RM ) 2) (Drug elimination) Suppose that sodium pentobarbitol is used to anesthetize a dog. The dog is anesthetized when its bloodstream contains at least 45mg of sodium pentobarbitol per kg of the dog s body weight. Suppose also that sodium pentobarbitol is eliminated exponentially from the dog s bloodstream, with a half-life of 5 hours. What single dose should be administered in order to anesthetize a 50-kg dog for 1 hour? (ANS: 2585 mg) 3) (Half-life Radioactive Decay) A breeder reactor converts relatively stable uranium 238 into the isotope plutonium 239. After 15 years, it is determined that 0.043% of the initial amount of plutonium has disintegrated. Find the half-life of this isotope if the rate of disintegration is proportional to the amount remaining. (ANS: years) 32

33 Electric Circuits - RC Given that the DE for an RL-circuit is Where is the voltage source is the inductance is the resistance Do you know what type of DE is this? CASE 1 : (constant) (1) i) Write in the linear equation form ii) Find the integrating factor, iii) Write down the equation as follows iv) Integrate the equation to find 33

34 CASE 2 : Consider or, the DE can be written as i) Write into the linear equation form and determine and ii) Find integrating factor, iii) Write down the equation with the integrating factor iv) Integrate the equation to find (1) Tabular Method Differentiate Sign Integrate

35 (2) From (1) (3) Replace (3) into (2) Exercise: 1) A 30-volt electromotive force is applied to an LR series circuit in which the inductance is 0.1 henry and the resistance is 15 ohms. Find the curve if. Determine the current as. 2) An electromotive force is applied to an LR series circuit in which the inductance is 20 henries and the resistance is 2 ohms. Find the current if. 35

36 Vertical Motion Newton s Second Law of Motion The Newton s Second Law of Motion is given by Where is the external force is the mass of the body is the velocity of the body with the same direction with is the time Example 1: A particle moves vertically under the force of gravity against air resistance, where is a constant. The velocity at any time is given by the differential equation If the particle starts off from rest, show that Such that. Then find the velocity as the time approaches infinity. Solution: i) Extract the information from the question Initial Condition ii) Separate the DE Let, 36

37 iii) Integrate the above equation Using Partial Fraction iv) Use the initial condition, v) Rearrange the equation 37

38 vi) Find the velocity as the time approaches infinity. When, => 38