1. 1 Engineering Mathematics Ⅰ 呂學育 博士 Oct. 6, 2004

2. 2 Short tangent segments suggest the shape of the curve 1.1.5 Direction Fields 輪廓 Slope= x y

3. 3 1.1.5 Direction Fields The general first-order differential equation has the form F(x, y, y’)=0 or in the explicit form y’=f(x,y) Note that, a graph of a solution of a first-order differential equation is called an integral curve of the equation. On the other hand, the slope of the integral curve through a given point (x 0,y 0 ) is y’(x 0 ).

4. 4 1.1.5 Direction Fields A drawing of the plane, with short line segments of slope drawn at selected points, is called a direction field of the differential equation. The name derives from the fact that at each point the line segment gives the direction of the integral curve through that point. The line segments are called lineal elements.

5. 5 1.1.5 Direction Fields Plotting Direction Fields – 1st Step y’=f(x,y)=C=constant  curves of equal inclination –2nd Step Along each curve f(x,y)=C, draw lineal elements  direction field –3rd Step Sketch approximate solution curves having the directions of the lineal elements as their tangent directions.

6. 6 1.2 Separable Equations A differential equation is called separable if it can be written as Such that we can separate the variables and write We attempt to integrate this equation

7. 7 1.2 Separable Equations Example 1. is separable. Write as Integrate this equation to obtain or in the explicit form What about y=0 ? Singular solution !

8. 8 1.2 Separable Equations Example 2. is separable, too. We write Integrate the separated equation to obtain The general solution is Again, check if y=-1 is a solution or not ?  it is a solution, but not a singular one, since it is a special case of the general solution 

9. 9 1.2.1 Some Applications of Separable Differential Equation Example 1.11 The Mathematical Policewoman Newton’s Law of Cooling From experimental observations it is known that (up to a ``satisfactory'' approximation) the time rate of change of the temperature of a body is proportional to the difference between the temperature of the body and the constant temperature of the surrounding medium. The differential equation that models this situation is where is the constant of proportionality.

10. 10 1.2.1 Some Applications of Separable Differential Equation Example 1.11 The Mathematical Policewoman http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/coobod.html#c1 Cooling of the Human Body

11. 11 Heat Transfer Mechanisms Heat transfer mechanisms can be grouped into 3 broad categories: Conduction – Conduction heat transfer is energy transport due to molecular motion and interaction. Conduction heat transfer through solids is due to molecular vibration. Fourier determined that Q/A, the heat transfer per unit area (W/m 2 ) is proportional to the temperature gradient dT/dx. The constant of proportionality is called the material thermal conductivity k Fouriers equation :

12. 12 Heat Transfer Mechanisms Heat transfer mechanisms can be grouped into 3 broad categories: Convection –Convection heat transfer is energy transport due to bulk fluid motion. Convection heat transfer through gases and liquids from a solid boundary results from the fluid motion along the surface. –Newton determined that the heat transfer/area, Q/A, is proportional to the fluid solid temperature difference Ts-Tf. The temperature difference usually occurs across a thin layer of fluid adjacent to the solid surface. This thin fluid layer is called a boundary layer. The constant of proportionality is called the heat transfer coefficient, h. –Newton's Equation:

13. 13 Heat Transfer Mechanisms Heat transfer mechanisms can be grouped into 3 broad categories: Radiation –Radiation heat transfer is energy transport due to emission of electromagnetic waves or photons from a surface or volume. The radiation does not require a heat transfer medium, and can occur in a vacuum. The heat transfer by radiation is proportional to the fourth power of the absolute material temperature T. The proportionality constant s is the Stefan-Boltzman constant equal to 5.67 x 10-8 W/m 2 K 4. The radiation heat transfer also depends on the material properties represented by, the emissivity of the material.

14. 14 1.2.1 Some Applications of Separable Differential Equation Example 1.11 The Mathematical Policewoman Determination of time of death. The lieutenant hands you the following report on a recent murder case. Police Report. Police arrived at the scene of a murder at 9:40 pm. They immediately took and recorded the temperature of the corpse, which was 94.4 ℉, and thoroughly inspected the area. By the time they finished the inspection, it was 11:00 pm. They again took the temperature of the corpse, which had dropped to 89.2 ℉, and had the corpse sent to the morgue. The temperature at the crime scene had remained steady at 68 ℉.

15. 15 1.2.1 Some Applications of Separable Differential Equation Example 1.11 The Mathematical Policewoman This is a separable (and linear) differential equation and its solution is solved as following: The constants and must be determined by using the initial conditions.

16. 16 1.2.1 Some Applications of Separable Differential Equation Example 1.11 The Mathematical Policewoman 1) set 9:40PM be time zero with the body temperature measured as 94.4 Thus far, 2) set at 11:00PM (80 min later) the body temperature is measured again and as 89.2, then 3) The temperature function of the body is 4) If the temperature at the time of death is 98.6, then the time of death is solved as min, i.e., the murder is at about 8:46 PM  

17. 17 1.3 Linear Differential Equations A first-order differential equation is linear if it has the form Multiply the differential equation by to get Now integrate to obtain The function is called an integrating factor for the differential equation. 

18. 18 1.3 Linear Differential Equations Linear: A differential equation is called linear if there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables. Non-linear: Differential equations that do not satisfy the definition of linear are non-linear. Quasi-linear: For a non-linear differential equation, if there are no multiplications among all dependent variables and their derivatives in the highest derivative term, the differential equation is considered to be quasi-linear.

19. 19 1.3 Linear Differential Equations Example 1.14 is a linear DE. P(x)=1 and q(x)=sin(x), both continuous for all x. An integrating factor is Multiply the DE by to get Or Integrate to get The general solution is

20. 20 1.3 Linear Differential Equations Example 1.15 Solve the initial value problem It can be written in linear form An integrating factor is for Multiply the DE by to get Or Integrate to get,then for For the initial condition, we need C=17/4  the solution of the initial value problem is

21. 21 1.4 Exact Differential Equations Definition1.3: Potential Function A function is a potential function for the differential equation on a region of the plane if, for each in, and Definition 1.4: Exact Differential Equation When a potential function exists on a region for the differential equation, then this equation is said to be exact on.

22. 22 1.4 Exact Differential Equations can be written in the form If there is a function such that and The differential equation becomes Which, by the chain rule, is the same as But it means that

23. 23 1.4 Exact Differential Equations can be written in the form If there is a function such that and (*) The differential equation becomes Which, by the chain rule, is the same as But it means that  implicitly defines a function y(x) that is general solution of the differential equation.  Thus, finding a function that satisfies equation(*) is equivalent to solving the differential equation. (note: verify by implicit differentiation)

24. 24 1.4 Exact Differential Equations Example 1.17 is neither separable nor linear. Write it in the form ($) It can in turn be written (*) Now let Observe that and Equation (*) becomes  The general solution of this equation is

25. 25 1.4 Exact Differential Equations Theorem 1.1 Test for Exactness is exact on if and only if, for each in,

26. 26 1.4 Exact Differential Equations If is exact, then there is a potential function and and Then, for (x,y) in R, Conversely, suppose and are continuous on R. Choose any (x 0,y 0 ) in R and define, for (x,y) in R, Immediately, we have

27. 27 1.4 Exact Differential Equations Next, compute And the proof is complete.

28. 28 1.4 Exact Differential Equations Example1.18 Here and and Note that, is satisfied by all (x,y) on a straight line. However, cannot hold for all (x,y) in an entire rectangle in the plane. Hence this differential equation is not exact on any rectangle.

29. 29 1.4 Exact Differential Equations Example1.19 Here and  for all (x,y) Therefore, this DE is exact. To find a potential function, set and

30. 30 1.4 Exact Differential Equations Example1.19 To find a potential function, set and Then, and we may choose The general sol of the DE is defined implicitly by