Chapter 38

1. Chapter 38 Differential Equations differential equations by Chtan (FYHS-Kulai)

2. What is a differential equation ? (D.E.) In short form differential equations by Chtan (FYHS-Kulai)

3. A differential equation is an equation that involves one or more derivatives, or differentials. differential equations by Chtan (FYHS-Kulai)

4. Differential equations play a prominent role in engineering, physics, economics, and other disciplines. differential equations by Chtan (FYHS-Kulai)

5. Differential equations are classified by 3 components : (a) type : ordinary or partial (b) order : the order of the highest-order derivative that occurs in the equation differential equations by Chtan (FYHS-Kulai)

6. (c)degree : the exponent of the highest power of the highest-order derivative, after the equation has been cleared of fractions and radicals in the dependent variable and its derivatives. differential equations by Chtan (FYHS-Kulai)

7. For example, is an ordinary differential equation, of order 3 and degree 2. differential equations by Chtan (FYHS-Kulai)

8. If the dependent variable y is a function of a single independent variable x, say y=f(x) Only “ordinary” derivatives occur ! differential equations by Chtan (FYHS-Kulai)

9. If the dependent variable y is a function of 2 or more independent variables, say if x and t are independent variables, then partial derivatives of y may occur. differential equations by Chtan (FYHS-Kulai)

10. For example, is a partial differential equation, of order 2 and degree one. [this is the 1-dimensional “wave-equation”.] differential equations by Chtan (FYHS-Kulai)

11. For a discussion of partial differential equations, including the wave equation and solutions of associated physical problems, see Kaplan, Advanced Calculus, Chapter 10. differential equations by Chtan (FYHS-Kulai)

12. Before we proceed to solve D.E., let us first examine how to form a D.E. from an ordinary equation including normal functions and trigonometric function or exponential function. differential equations by Chtan (FYHS-Kulai)

13. Let us see the following equation, Then, This is an D.E. differential equations by Chtan (FYHS-Kulai)

14. e.g. 1 Eliminate the arbitrary constant A from the equation : differential equations by Chtan (FYHS-Kulai)

15. Soln : Integrating both sides w.r.t. x differential equations by Chtan (FYHS-Kulai)

16. differential equations by Chtan (FYHS-Kulai)

17. e.g. 2 Eliminate the arbitrary constants A and B from the equation : differential equations by Chtan (FYHS-Kulai)

18. Soln : differential equations by Chtan (FYHS-Kulai)

19. Do p250 and p251 Ex 19a differential equations by Chtan (FYHS-Kulai)

20. We willfocus on ordinary differential equations(no partial D.E.). In outline, these are the things we will study : differential equations by Chtan (FYHS-Kulai)

21. First-order equations • (a) variables separable • (b) homogeneous • (c) linear • 2. Special types of second-order equations differential equations by Chtan (FYHS-Kulai)

22. 2nd order Linear equations with constant coefficients • (a) homogeneous • (b) nonhomogeneous differential equations by Chtan (FYHS-Kulai)

23. A First-order equations with variables separable differential equations by Chtan (FYHS-Kulai)

24. A first-order differential equation can be solved by integration if it is possible to collect all y terms with dy and all x terms with dx. It is possible to write the equation in the form differential equations by Chtan (FYHS-Kulai)

25. Then the general solution (G.S.) is : where C is an arbitrary constant. differential equations by Chtan (FYHS-Kulai)

26. e.g. 3 Solve the equation differential equations by Chtan (FYHS-Kulai)

27. Soln : differential equations by Chtan (FYHS-Kulai)

28. Do p. 253 Ex 19b differential equations by Chtan (FYHS-Kulai)

29. B First-order homogeneous equations differential equations by Chtan (FYHS-Kulai)

30. A differential equation that can be put into the form is said to be homogeneous. Such an equation can be solved by introducing a new dependent variable differential equations by Chtan (FYHS-Kulai)

31. Then , Solved by separation of variables : differential equations by Chtan (FYHS-Kulai)

32. e.g. 4 Show that the equation is homogeneous, and solve it. differential equations by Chtan (FYHS-Kulai)

33. Soln : From the given equation, we have differential equations by Chtan (FYHS-Kulai)

34. The solution of this is : differential equations by Chtan (FYHS-Kulai)

35. differential equations by Chtan (FYHS-Kulai)

36. Do p. 255 Ex 19c differential equations by Chtan (FYHS-Kulai)

37. C First-order linear equations differential equations by Chtan (FYHS-Kulai)

38. is of the first degree is of the second degree If every term of a differential equation is of degree zero or degree one, then the equation is linear. differential equations by Chtan (FYHS-Kulai)

39. A linear differential equation of first order can always be put into the standard form : where P and Q are functions of x. differential equations by Chtan (FYHS-Kulai)

40. * 1st order Liner equations are solved by multiplying throughout by the function : is known as an integrating factor. differential equations by Chtan (FYHS-Kulai)

41. differential equations by Chtan (FYHS-Kulai)

42. differential equations by Chtan (FYHS-Kulai)

43. e.g. 5 Solve the equation differential equations by Chtan (FYHS-Kulai)

44. Soln : This is of the form with P=-1,Q=x Now, multiplying both sides by differential equations by Chtan (FYHS-Kulai)

45. differential equations by Chtan (FYHS-Kulai)

46. Integration by parts Multiply both sides by differential equations by Chtan (FYHS-Kulai)

47. e.g. 6 If and Find y in terms of x. differential equations by Chtan (FYHS-Kulai)

48. Soln : The equation is of the form differential equations by Chtan (FYHS-Kulai)

49. The G.S. is : differential equations by Chtan (FYHS-Kulai)

50. differential equations by Chtan (FYHS-Kulai)