8 สมการเช งอน พ นธ และ การประย กต

Transcription

1 General Mathematics General Mathematics For the students from Pharmaceutical Faculty 1/2004 Instructor: Dr Wattana Toutip (ดร.ว ฒนา เถาว ท พย ) 8 สมการเช งอน พ นธ และ การประย กต (Differential Equation andapplications) Overview: This chapter shows how to solve the first ordered differential equations. Some applications will be discussed in the last section of the chapter. 8.1 First Ordered Differential Equations Definition An equation containing the derivatives or differential of one dependent variable with respect to one independent variable is said to be an ordinary differential equation. For eample, 2y 3 d (1.1) 2 d y 3 y 0 2 d d (1.2) ( y) d 2 3 (1.3) y y 4 (1.4) y y 3 (1.5) The order of the highest derivative in a differential equation is called the order of the equation. For eample, Equation (1.1), (1.3) and (1.4) is the first order differential equations. Equation (1.2) and (1.5) are the second order differential equations. In this course, we discuss only the first order differential equations.

2 General Mathematics 2 Eample 1. Show that y 3e is a solution of the initial value problem y y 0 Eample 2. Show that the following functions,, 1. y 3e 2. y 4e 3. y Ce are solutions of the differential equation y y 0 A solution of a differential equation that contains arbitrary constant is called the general solution. A solution that is free of arbitrary constant is called the particular solution. For eamples, from Eample 2, y Ce is the general solution. y 3e and y 4e are the particular solutions.

3 General Mathematics 3 We are often interested in solving the first-order differential equation f (, y) d (1.6) subject to the condition y( 0) y0, where 0 is a number in an interval. The problem Solve : f (, y) d (1.7) Subject to : y( 0) y0 is called an Initial Value Problem. The condition is known as Initial condition. Eample 4. Show that y 3e is a particular solution of the initial value problem y y, and y(0) Eample 5. Show that y C is a general solution of the differential equation 2yy How to find solutions of the given differential equations. We are going to discuss this problem in the following sections.

4 General Mathematics Variable Separable Equations We begin our stu of the method of solving first-order differential equations with the simplest of all differential equations. If f( ) is a given continuous function, then the first-order differential equation f( ) d (1.8) can be solved by integration. The solution of (1.8) is y f ( ) d C. Eample 6. Solve 2 1 2e d. Eample 7. Solve sin d. Definition A differential equation of the form f ( ) d g( y) is said to be separable or to have separable variables. We can see that if y h( ), then g( h( )) d( h( )) f ( ) d g( h( )) h ( ) d f ( ) d g y f d g( h( )) h ( ) d f ( ) d Therefore, ( ) ( )

5 General Mathematics 5 Eample 8. Solve d y. 1 Eample 9. Solve d. y y Eample 10. Solve e sin d y 0.

6 General Mathematics Homogeneous Equations First, we need the definition of homogeneous functions. Definition If f ( t, ty) t n f (, y) for some real number n, then f (, y) is said to be a homogeneous function of degree n. For eamples, Eample 11. Verify that the following functions are homogeneous or not. 2 a) f (, y) y 3 b) c) f y y 2 2 (, ) 3 4 f (, y) y 3 2 We can see from Eample 11 that f (, y ) will be a homogeneous function if the total degrees of each term are the same. For eamples, a) b) c) f (, y) 6y 3 y is homogeneous of degree 5. f (, y) 6y 3 y is not homogeneous. f (, y) 6y 3 y is not homogeneous.

7 General Mathematics 7 Note that if f (, y ) is a homogeneous function of degree n, then we can write y n f (, y) f (1, u), where y u or u and f (, y) y n f ( v,1), where vy or v. y Eample 12. Given f (, y) 2y y 2 2. Write (, ) n f (, y) f (1, u), where y u or f y in the form y u Method for solution using homogeneous function An equation of the form M(, y) d N(, y) 0, (1.9) where M (, y ) and N(, y) are the same degree homogeneity, can be u or vy reduced to separable variable by substitution y follow: If y u, then ud du. Substituting in (1.9) we obtain (for y u ) as n n M(1, u) d N(1, u)( ud du) 0 (1.10) or M(1, u) d N(1, u)( ud du) 0 (1.11)

8 General Mathematics 8 which gives d (1, ) N u du M (1, u) un(1, u) 0 (1.12) We can see that Equation (1.12) is in the form of separable variables between and the new one u, and can be solved by that method. Eample 13. Solve ( y ) d ( y) 0 Eample 14. Solve yd ( y ) 0

9 General Mathematics The First Ordered Linear Differential Equations The first order differential equation of the form a( ) b( ) y c( ) d where a ( ) 0, is called first order linear differential equation. For eamples 2 2y d 2 3y 4 d, etc. How to solve the linear differential equations

10 General Mathematics 10 Summary of the method 1. Make the equation in the form p( ) y f ( ) d p( ) d 2. Find the integrating factor e 3. Multiply the equation by the integrating factor 4. Rearrange the equation in the form d p( ) d p( ) d [ ye ] f ( ) e d 5. Integrate both side of the equation in step 4. Eample 15. Solve 3y 0 d Eample 16. Solve d 6 4y e

11 General Mathematics 11 Eample 17. Solve y 2 d, subject to y(1) 0. Eample 18. Solve 2y d, subject to y(0) 3.

12 General Mathematics Applications of Differential Equations I. Growth and Decay II. Cooling and Chemical Miture Growth and Decay Problems occur in physical theories involving either growth and decay. For eample, in biology it is often observed that the rate at which bacteria grow is proportional to the number of bacteria present. Suppose that at the time t the number of bacteria is N(t). Since the rate of growth is proportional to the number of bacteria, then we have the equation d( N( t)) dt k, (1.13) Nt () for some constant k. We can see that Equation (1.13) is a first order linear differential equation as d( N( t)) dt k. N( t) (1.14) We can solve Equation (1.14) as discussed in the previous section.

13 General Mathematics 13 Eample 1. A culture initially has N0 of bacterial. At time t 1 hour the 3 number of bacterial is measured to be 0 2 N. If the rate of growth is proportional to the number of bacteria present, determine the time for the number of bacterial to be triple.

14 General Mathematics 14 Eample 2. Suppose that the half-life of the radioactive C-14 is 1 approimately 5600 years. A bone is found to contain the original 1000 amount of C-14. Determine the age of the fossil.

15 General Mathematics 15 Cooling and Chemical Miture Newton s law of cooling states that the rate at which the temperature Ttchange () in a cooling bo is proportional to the difference between the temperature in the bo and the constant temperature T 0 of the surrounding medium that is d( T( t)) k( T( t) T0 ) (1.15) dt where k is a constant of proportionality. Eample 3. When the cake is removed from a baking oven, its temperature is measured at 300 F. Three minutes later its temperature is 200 F. How long will it take to cool off to the temperature 80 F if a room temperature is 70 F?

16 General Mathematics 16 The miing of two fluids sometimes gives rise to a linear first order differential equation. In the net eample, we consider the miture of two salt solutions of different concentrations. Eample 4. Initially 50 pounds of salt is dissolved in a large tank holding 300 gallons of water. A brine solution is pumped into the tank at a rate of 3 gallons per minute, and the well-stirred solution is then pumped out at the same rate. If the concentration of the solution entering is 2 pounds per gallon, determine the amount of salt in the tank at any time. How much salt is present after along time?