1. EUT 102 1 CHAPTER 2 : First Order Differential Equations

2. EUT 1022 Learning Outcomes a) At the end of this chapter, it is expected that all students will be able to know: What is ODE? What is ODE? What is order of ODE? What is order of ODE? How to solve ODE? How to solve ODE? How to use IVP and BVP? How to use IVP and BVP? b) And also at the end of the chapter students will have to possess the skills of solving first-order ODE.

3. EUT 102 3 An equation that contains an unknown function and one or more of its derivatives WHAT IS A DIFFERENTIAL EQUATION?

4. EUT 102 4 ODE BASICS “An ordinary differential equation (ODE) is an equation that contains one or several derivatives of an unknown function.” (Kreyzig). For example:

5. EUT 102 5 Order of a Differential Equation The order of a differential equation is the order of the highest derivative that appears in the equation. First order differential equations can be written as:

6. EUT 102 6 Example 2.1

7. EUT 102 7 Solution of an ODE A function f is called a solution of a differential equation if the equation is satisfied when y=f(x) and its derivatives are substituted into the equation.

8. EUT 102 8 Example Solve the following differential equations :

9. EUT 102 9 FIRST ORDER DIFFERENTIAL EQUATIONS Separable Equations Homogeneous Equations Linear Equations Exact Equations Applications

10. EUT 102 10 2.1 : SEPARABLE EQUATIONS The differential equation ; y’ = f(x,y) is said to be separable if the equation can be written as the product of a function of x, u(x) and a function of y,v(y) i.e. y’ = u(x)v(y)

11. EUT 102 11 Example 2.2 y’ - xy = x can be written as y’ = x(1+y) i.e u(x) = x & v(y) = 1+y y’ siny cosx – cosy sinx = 0 or can be written as y’ = tan x cot y

12. EUT 102 12 Exercise Determine each of the following ODEs is separable equations or not. a) y’ = xe y-x b) y’x = x – 2y

13. EUT 102 13 Solution of Separable Equations The equation y’ = u(x)v(y) can be written in the form ; Then, we integrate on both sides of the equation: where A is constant

14. EUT 102 14 Example 2.3 Solve the following differential equations : a)(x + 2)y’ = y b) y’e x + xy 2 = 0 c)x 2 y’ = 1 + y

15. EUT 102 15 Solutions a) y -1 dy = (x+2) -1 dx By integrating both sides : ln |y| = ln |x+2| + A ln |y| - ln |x+2| = A and becomes y = B(x+2).

16. EUT 102 16 Example 2.4 Let θ be the temperature (in o C) for a mass in a room with a constant room temperature at 18 o C. The mass cools from 70 o C to 57 o C in 5 minutes, how much longer will be needed by the mass to cool down to 40 o C.

17. EUT 102 17 Solution Hint…

18. EUT 102 18 = 18 + Ae -kt – From the initial condition, we obtain A = 52 and k = 0.05753. – Solve … The mass will need approximately 9.952 minutes more to cool.

19. EUT 102 19 2.2 : HOMOGENEOUS EQUATIONS Definition 2.2 : A differential equation y’ = f(x,y) is said to be homogeneous if f(λx, λy) = f(x, y) for any real number λ.

20. EUT 102 20 Example 2.5 Determine whether y’y = x(lny – lnx) is a homogeneous equation.

21. EUT 102 21 Solution Then, show that f(λx, λy) = f(x,y).

22. EUT 102 22 Example 2.6

23. EUT 102 23 Solution of Homogeneous Equations A homogeneous equation can be transformed into separable equations by substituting y = xv or v = y/x. Hence, y’ = v + x (dv/dx) This will result in separable equations in variables x and v. Solve and resubstitute the value v=y/x.

24. EUT 102 24 Example 2.7 Solve the initial value problem. y(0) = 2.

25. EUT 102 25 Solution Test the homogenity Substitute y = xv Integrate

26. EUT 102 26 Substitute v = y/x and determine the general solution. The initial condition, y(0) = 2, so that A = 2.

27. EUT 102 27 Sekalipun mereka dapat memahami hakikat ‘bagaimana hendak bermula tetapi tidak pernah bersiap’…. Sampai bila pun mereka tidak akan bermula “Peluang sentiasa ada kepada setiap orang. Hanya kesanggupan yang membezakan”

28. EUT 102 28 Example 2.8

29. EUT 102 29

30. EUT 102 30 2.3 : LINEAR EQUATIONS Definition 2.3 : (First OLDEs) An equation is said to be a first-order linear equation if it has the form a(x)y’ + b(x)y = c(x) where a(x), b(x) & c(x) are continuous functions on a given interval.(or a constant).

31. EUT 102 31 Example 2.9 a) xy’ – 2y = x + 1 is a linear equation with a(x) = x, b(x) = -2 dan c(x) = x+1 b) 2x 2 y’ + xe y = sin x is not a linear equation.

32. EUT 102 32 Example 2.10 Determine whether the following ODEs are linear equations or not. a) (1 - x 2 )y’ = x(y + sin -1 x) b) y’ + xy 2 = e x

33. EUT 102 33 Solution of Linear Equations We rewrite the linear the equation as y’ + p(x)y = q(x) a(x) = 1 find ∫p(x)dx An integrating factor is ρ = e ∫p(x)dx

34. EUT 102 34 Cont..... multiplying both sides of the linear equation by ρ we get, ρy ’ + ρp(x)y = ρq(x) Or can be rewrite into the form; The solution to the linear equation is by integrating both sides with respect to x.

35. EUT 102 35 Example 2.11 Solve the following linear equations. a) y’ + y = x b) y’ +y tan x = kos x, given that y = 1 when x = 0.

36. EUT 102 36 Solutions a) y’ + y = x we know that p(x) = 1 and q(x) = x. Answer : y = (x – 1) + Ae -x b) Answer : y = (x + 1) kos x

37. EUT 102 37 Example 2.12: Linear Equations Application A tank containing 50 liters of liquid with composition, 90% water and 10% alcohol. A second liquid with composition, 50% water and 50% alcohol is poured into the tank at the rate of 4 liters per minute. While the second liquid is poured into the tank, the liquid in the tank is drained out at the rate of 5 liters per minute. Assuming that the liquid in the tank mix uniformly, how much alcohol left in the tank 10 minutes later?

38. EUT 102 38 Answer 13.45 liter

39. EUT 102 39 2.4 : EXACT EQUATIONS Definition 2.4 A first order differential equation of the form M(x,y)dx + N(x,y)dy = 0 is called an exact diffrential equation if the differential form M(x,y)dx + N(x,y)dy is exact,that is, du = M(x,y)dx + N(x,y)dy of some function u(x,y).

40. EUT 102 40 Theorem 2.1 : (Condition of an exact equation) * The equation M(x,y)dx + N(x,y)dy = 0 is an exact equation if and only if :

41. EUT 102 41 Example 2.13 Show that (6xy+2y)y’ = –(2x + 3y 2 ) is an exact equation.

42. EUT 102 42 Solution We rewrite in the form (2x + 3y 2 )dx + (6xy + 2y)dy = 0 i.e. M = (2x + 3y 2 ) and N = (6xy + 2y) Then, we test for exactness

43. EUT 102 43 Example 2.14 Solve the following differential equation; (1- kos 2 x) dy + (y sin 2x) dx = 0. Solution :

44. EUT 102 44 Solution of An Exact Equations 1) Write in the form of exact equation : M(x,y)dx + N(x,y)dy = 0. Test for Exactness :

45. EUT 102 45 Cont… 2) Write

46. EUT 102 46 Cont.... 3) By integrating Equation (I) with respect to x, we have u(x,y) = ∫ M dx + Φ(y). 4) Differentiate u with respect to y, and equating the result with equation (II) to determine the function Φ(y). 5) Write the solution in the form of u(x,y) = A, where A is a constant. 5) By substituting the initial condition, we will get the particular solution of the initial value problem

47. EUT 102 47 Example 2.14 Show the the following ODE is an exact equation and solve the differential equation. (6x 2 - 10xy + 3y 2 )dx + (-5x 2 + 6xy -3y 2 )dy = 0

48. EUT 102 48 Solution Test for Exactness (6x 2 - 10xy + 3y 2 ) = M and (-5x 2 + 6xy -3y 2 ) = N Show that

49. EUT 102 49 Cont … Let u(x,y) be the solution, then u = ∫Mdx +Φ(y) u = ∫(6x 2 - 10xy + 3y 2 )dx + Φ(y) = 2x 3 – 5x 2 y +3xy 2 + Φ(y) differentiate u with respect to y, we obtain ∂u/∂y = -5x 2 +6xy + Φ ’ (y)

50. EUT 102 50 Cont..... By equating with N we obtain, -5x 2 +6xy + Φ ’ (y) = (-5x 2 + 6xy -3y 2 ) hence; Φ ’ (y) = -3y 2 and we obtain Φ(y) = -y 3 + B The general solution is u(x,y) = A, where u(x,y) = 2x 3 – 5x 2 y +3xy 2 - y 3 + B = A u(x,y) = 2x 3 – 5x 2 y +3xy 2 - y 3 = C, C = A - B

51. EUT 102 51 Example 2.15 Solve the following differential equations 1) xy’ + y + 4 = 0 2) sin x dy + (y kos x – x sin x) dx = 0

52. EUT 102 52 Answer 1) u(x,y) = (y + 4)x = C, C = A – B 2) u(x,y) = y sin x + x kos x – sin x = C, C = A – B

53. EUT 102 53 Exercise 2 Show that the following differential equation 2y dx + x dy = 0 is not an exact equation. If the integrating factor, solve the differential equation.

54. EUT 102 54 Answer x 2 y = A, A is a constant